Excitation and use of guided surface waves

ABSTRACT

Disclosed are various embodiments for transmitting and receiving energy conveyed in the form of a guided surface-waveguide mode along the surface of a lossy medium such as, e.g., a terrestrial medium excited by a guided surface waveguide probe.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/728,492, entitled “EXCITATION AND USE OF GUIDED SURFACE WAVES,” filedJun. 2, 2015, the entire contents of both of which applications arehereby incorporated herein by reference.

This application is also related to U.S. Non-provisional patentapplication Ser. No. 13/789,538, entitled “Excitation and Use of GuidedSurface Wave Modes on Lossy Media,” which was filed on Mar. 7, 2013, theentire contents of which is hereby incorporated herein by reference inits entirety. This application is also related to U.S. Non-provisionalpatent application Ser. No. 13/789,525, entitled “Excitation and Use ofGuided Surface Wave Modes on Lossy Media,” which was filed on Mar. 7,2013, the entire contents of which is hereby incorporated herein byreference in its entirety. This application is also related to U.S.Non-provisional patent application Ser. No. 14/483,089, entitled“Excitation and Use of Guided Surface Wave Modes on Lossy Media,” whichwas filed on Sep. 10, 2014, the entire contents of which is herebyincorporated herein by reference in its entirety. This application isalso related to U.S. Non-provisional patent application Ser. No.14/728,507, entitled “Excitation and Use of Guided Surface Waves,” whichwas filed on Jun. 2, 2015, the entire contents of which is herebyincorporated herein by reference in its entirety.

BACKGROUND

For over a century, signals transmitted by radio waves involvedradiation fields launched using conventional antenna structures. Incontrast to radio science, electrical power distribution systems in thelast century involved the transmission of energy guided along electricalconductors. This understanding of the distinction between radiofrequency (RF) and power transmission has existed since the early1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 is a chart that depicts field strength as a function of distancefor a guided electromagnetic field and a radiated electromagnetic field.

FIG. 2 is a drawing that illustrates a propagation interface with tworegions employed for transmission of a guided surface wave according tovarious embodiments of the present disclosure.

FIG. 3 is a drawing that illustrates a guided surface waveguide probedisposed with respect to a propagation interface of FIG. 2 according toan embodiment of the present disclosure.

FIG. 4 is a plot of an example of the magnitudes of close-in and far-outasymptotes of first order Hankel functions according to variousembodiments of the present disclosure.

FIGS. 5A and 5B are drawings that illustrate a complex angle ofincidence of an electric field synthesized by a guided surface waveguideprobe according to the various embodiments of the present disclosure.

FIG. 6 is a graphical representation illustrating the effect ofelevation of a charge terminal on the location where the electric fieldof FIG. 5A intersects with the lossy conducting medium at a Brewsterangle according to various embodiments of the present disclosure.

FIG. 7 is a graphical representation of an example of a guided surfacewaveguide probe according to an embodiment of the present disclosure.

FIGS. 8A through 8C are graphical representations illustrating examplesof equivalent image plane models of the guided surface waveguide probeof FIGS. 3 and 7 according to various embodiments of the presentdisclosure.

FIGS. 9A and 9B are graphical representations illustrating examples ofsingle-wire transmission line and classic transmission line models ofthe equivalent image plane models of FIGS. 8B and 8C according tovarious embodiments of the present disclosure.

FIG. 10 is a flow chart illustrating an example of adjusting a guidedsurface waveguide probe of FIGS. 3 and 7 to launch a guided surface wavealong the surface of a lossy conducting medium according to variousembodiments of the present disclosure.

FIG. 11 is a plot illustrating an example of the relationship between awave tilt angle and the phase delay of a guided surface waveguide probeof FIGS. 3 and 7 according to various embodiments of the presentdisclosure.

FIG. 12 is a Smith chart illustrating an example of adjusting the loadimpedance of the guided surface waveguide probe of FIGS. 3 and 7according to various embodiments of the present disclosure.

FIG. 13 is a plot comparing measured and theoretical field strength ofthe guided surface waveguide probe of FIGS. 3 and 7 according to anembodiment of the present disclosure.

FIGS. 14A through 14C depict examples of receiving structures that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIG. 14D is a flow chart illustrating an example of adjusting areceiving structure according to various embodiments of the presentdisclosure.

FIG. 15 depicts an example of an additional receiving structure that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIG. 16A depicts a schematic diagram representing theThevenin-equivalent of the receivers depicted in FIGS. 14A and 14Baccording to an embodiment of the present disclosure.

FIG. 16B depicts a schematic diagram representing the Norton-equivalentof the receiver depicted in FIG. 15 according to an embodiment of thepresent disclosure.

FIGS. 17A and 17B are schematic diagrams representing examples of aconductivity measurement probe and an open wire line probe,respectively, according to an embodiment of the present disclosure.

FIG. 18 is a schematic drawing of an example of an adaptive controlsystem employed by the probe control system of FIG. 3 according tovarious embodiments of the present disclosure.

FIGS. 19A-19B and 20 are drawings of examples of variable terminals foruse as a charging terminal according to various embodiments of thepresent disclosure.

DETAILED DESCRIPTION

To begin, some terminology shall be established to provide clarity inthe discussion of concepts to follow. First, as contemplated herein, aformal distinction is drawn between radiated electromagnetic fields andguided electromagnetic fields.

As contemplated herein, a radiated electromagnetic field compriseselectromagnetic energy that is emitted from a source structure in theform of waves that are not bound to a waveguide. For example, a radiatedelectromagnetic field is generally a field that leaves an electricstructure such as an antenna and propagates through the atmosphere orother medium and is not bound to any waveguide structure. Once radiatedelectromagnetic waves leave an electric structure such as an antenna,they continue to propagate in the medium of propagation (such as air)independent of their source until they dissipate regardless of whetherthe source continues to operate. Once electromagnetic waves areradiated, they are not recoverable unless intercepted, and, if notintercepted, the energy inherent in the radiated electromagnetic wavesis lost forever. Electrical structures such as antennas are designed toradiate electromagnetic fields by maximizing the ratio of the radiationresistance to the structure loss resistance. Radiated energy spreads outin space and is lost regardless of whether a receiver is present. Theenergy density of the radiated fields is a function of distance due togeometric spreading. Accordingly, the term “radiate” in all its forms asused herein refers to this form of electromagnetic propagation.

A guided electromagnetic field is a propagating electromagnetic wavewhose energy is concentrated within or near boundaries between mediahaving different electromagnetic properties. In this sense, a guidedelectromagnetic field is one that is bound to a waveguide and may becharacterized as being conveyed by the current flowing in the waveguide.If there is no load to receive and/or dissipate the energy conveyed in aguided electromagnetic wave, then no energy is lost except for thatdissipated in the conductivity of the guiding medium. Stated anotherway, if there is no load for a guided electromagnetic wave, then noenergy is consumed. Thus, a generator or other source generating aguided electromagnetic field does not deliver real power unless aresistive load is present. To this end, such a generator or other sourceessentially runs idle until a load is presented. This is akin to runninga generator to generate a 60 Hertz electromagnetic wave that istransmitted over power lines where there is no electrical load. Itshould be noted that a guided electromagnetic field or wave is theequivalent to what is termed a “transmission line mode.” This contrastswith radiated electromagnetic waves in which real power is supplied atall times in order to generate radiated waves. Unlike radiatedelectromagnetic waves, guided electromagnetic energy does not continueto propagate along a finite length waveguide after the energy source isturned off. Accordingly, the term “guide” in all its forms as usedherein refers to this transmission mode of electromagnetic propagation.

Referring now to FIG. 1, shown is a graph 100 of field strength indecibels (dB) above an arbitrary reference in volts per meter as afunction of distance in kilometers on a log-dB plot to furtherillustrate the distinction between radiated and guided electromagneticfields. The graph 100 of FIG. 1 depicts a guided field strength curve103 that shows the field strength of a guided electromagnetic field as afunction of distance. This guided field strength curve 103 isessentially the same as a transmission line mode. Also, the graph 100 ofFIG. 1 depicts a radiated field strength curve 106 that shows the fieldstrength of a radiated electromagnetic field as a function of distance.

Of interest are the shapes of the curves 103 and 106 for guided wave andfor radiation propagation, respectively. The radiated field strengthcurve 106 falls off geometrically (1/d, where d is distance), which isdepicted as a straight line on the log-log scale. The guided fieldstrength curve 103, on the other hand, has a characteristic exponentialdecay of e^(−αd)/√{square root over (d)} and exhibits a distinctive knee109 on the log-log scale. The guided field strength curve 103 and theradiated field strength curve 106 intersect at point 113, which occursat a crossing distance. At distances less than the crossing distance atintersection point 113, the field strength of a guided electromagneticfield is significantly greater at most locations than the field strengthof a radiated electromagnetic field. At distances greater than thecrossing distance, the opposite is true. Thus, the guided and radiatedfield strength curves 103 and 106 further illustrate the fundamentalpropagation difference between guided and radiated electromagneticfields. For an informal discussion of the difference between guided andradiated electromagnetic fields, reference is made to Milligan, T.,Modern Antenna Design, McGraw-Hill, 1^(st) Edition, 1985, pp. 8-9, whichis incorporated herein by reference in its entirety.

The distinction between radiated and guided electromagnetic waves, madeabove, is readily expressed formally and placed on a rigorous basis.That two such diverse solutions could emerge from one and the samelinear partial differential equation, the wave equation, analyticallyfollows from the boundary conditions imposed on the problem. The Greenfunction for the wave equation, itself, contains the distinction betweenthe nature of radiation and guided waves.

In empty space, the wave equation is a differential operator whoseeigenfunctions possess a continuous spectrum of eigenvalues on thecomplex wave-number plane. This transverse electro-magnetic (TEM) fieldis called the radiation field, and those propagating fields are called“Hertzian waves.” However, in the presence of a conducting boundary, thewave equation plus boundary conditions mathematically lead to a spectralrepresentation of wave-numbers composed of a continuous spectrum plus asum of discrete spectra. To this end, reference is made to Sommerfeld,A., “Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie,”Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A.,“Problems of Radio,” published as Chapter 6 in Partial DifferentialEquations in Physics—Lectures on Theoretical Physics: Volume VI,Academic Press, 1949, pp. 236-289, 295-296; Collin, R. E., “HertzianDipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20^(th)Century Controversies,” IEEE Antennas and Propagation Magazine, Vol. 46,No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P. F, Krauss,H. L., and Skalnik, J. G., Microwave Theory and Techniques, VanNostrand, 1953, pp. 291-293, each of these references being incorporatedherein by reference in their entirety.

The terms “ground wave” and “surface wave” identify two distinctlydifferent physical propagation phenomena. A surface wave arisesanalytically from a distinct pole yielding a discrete component in theplane wave spectrum. See, e.g., “The Excitation of Plane Surface Waves”by Cullen, A. L., (Proceedings of the IEE (British), Vol. 101, Part IV,August 1954, pp. 225-235). In this context, a surface wave is consideredto be a guided surface wave. The surface wave (in the Zenneck-Sommerfeldguided wave sense) is, physically and mathematically, not the same asthe ground wave (in the Weyl-Norton-FCC sense) that is now so familiarfrom radio broadcasting. These two propagation mechanisms arise from theexcitation of different types of eigenvalue spectra (continuum ordiscrete) on the complex plane. The field strength of the guided surfacewave decays exponentially with distance as illustrated by curve 103 ofFIG. 1 (much like propagation in a lossy waveguide) and resemblespropagation in a radial transmission line, as opposed to the classicalHertzian radiation of the ground wave, which propagates spherically,possesses a continuum of eigenvalues, falls off geometrically asillustrated by curve 106 of FIG. 1, and results from branch-cutintegrals. As experimentally demonstrated by C. R. Burrows in “TheSurface Wave in Radio Propagation over Plane Earth” (Proceedings of theIRE, Vol. 25, No. 2, February, 1937, pp. 219-229) and “The Surface Wavein Radio Transmission” (Bell Laboratories Record, Vol. 15, June 1937,pp. 321-324), vertical antennas radiate ground waves but do not launchguided surface waves.

To summarize the above, first, the continuous part of the wave-numbereigenvalue spectrum, corresponding to branch-cut integrals, produces theradiation field, and second, the discrete spectra, and correspondingresidue sum arising from the poles enclosed by the contour ofintegration, result in non-TEM traveling surface waves that areexponentially damped in the direction transverse to the propagation.Such surface waves are guided transmission line modes. For furtherexplanation, reference is made to Friedman, B., Principles andTechniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286,290, 298-300.

In free space, antennas excite the continuum eigenvalues of the waveequation, which is a radiation field, where the outwardly propagating RFenergy with E_(Z) and H_(ϕ) in-phase is lost forever. On the other hand,waveguide probes excite discrete eigenvalues, which results intransmission line propagation. See Collin, R. E., Field Theory of GuidedWaves, McGraw-Hill, 1960, pp. 453, 474-477. While such theoreticalanalyses have held out the hypothetical possibility of launching opensurface guided waves over planar or spherical surfaces of lossy,homogeneous media, for more than a century no known structures in theengineering arts have existed for accomplishing this with any practicalefficiency. Unfortunately, since it emerged in the early 1900's, thetheoretical analysis set forth above has essentially remained a theoryand there have been no known structures for practically accomplishingthe launching of open surface guided waves over planar or sphericalsurfaces of lossy, homogeneous media.

According to the various embodiments of the present disclosure, variousguided surface waveguide probes are described that are configured toexcite electric fields that couple into a guided surface waveguide modealong the surface of a lossy conducting medium. Such guidedelectromagnetic fields are substantially mode-matched in magnitude andphase to a guided surface wave mode on the surface of the lossyconducting medium. Such a guided surface wave mode can also be termed aZenneck waveguide mode. By virtue of the fact that the resultant fieldsexcited by the guided surface waveguide probes described herein aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium, a guided electromagnetic fieldin the form of a guided surface wave is launched along the surface ofthe lossy conducting medium. According to one embodiment, the lossyconducting medium comprises a terrestrial medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides foran examination of the boundary value solutions to Maxwell's equationsderived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck,J., “On the Propagation of Plane Electromagnetic Waves Along a FlatConducting Surface and their Relation to Wireless Telegraphy,” Annalender Physik, Serial 4, Vol. 23, Sep. 20, 1907, pp. 846-866. FIG. 2depicts cylindrical coordinates for radially propagating waves along theinterface between a lossy conducting medium specified as Region 1 and aninsulator specified as Region 2. Region 1 can comprise, for example, anylossy conducting medium. In one example, such a lossy conducting mediumcan comprise a terrestrial medium such as the Earth or other medium.Region 2 is a second medium that shares a boundary interface with Region1 and has different constitutive parameters relative to Region 1. Region2 can comprise, for example, any insulator such as the atmosphere orother medium. The reflection coefficient for such a boundary interfacegoes to zero only for incidence at a complex Brewster angle. SeeStratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941, p. 516.

According to various embodiments, the present disclosure sets forthvarious guided surface waveguide probes that generate electromagneticfields that are substantially mode-matched to a guided surface waveguidemode on the surface of the lossy conducting medium comprising Region 1.According to various embodiments, such electromagnetic fieldssubstantially synthesize a wave front incident at a complex Brewsterangle of the lossy conducting medium that can result in zero reflection.

To explain further, in Region 2, where an e^(jωt) field variation isassumed and where ρ≠0 and z≥0 (with z being the vertical coordinatenormal to the surface of Region 1, and ρ being the radial dimension incylindrical coordinates), Zenneck's closed-form exact solution ofMaxwell's equations satisfying the boundary conditions along theinterface are expressed by the following electric field and magneticfield components:

$\begin{matrix}{{H_{2\varphi} = {{Ae}^{{- u_{2}}z}\mspace{14mu} {H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},} & (1) \\{{E_{2\rho} = {{A\left( \frac{u_{2}}{j\; {\omega ɛ}_{o}} \right)}e^{{- u_{2}}z}\mspace{14mu} {H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},{and}} & (2) \\{E_{2z} = {{A\left( \frac{- \gamma}{{\omega ɛ}_{o}} \right)}e^{{- u_{2}}z}\mspace{14mu} H_{0}^{(2)}{\left( {{- j}\; {\gamma\rho}} \right).}}} & (3)\end{matrix}$

In Region 1, where the e^(jωt) field variation is assumed and where ρ≠0and z≤0, Zenneck's closed-form exact solution of Maxwell's equationssatisfying the boundary conditions along the interface is expressed bythe following electric field and magnetic field components:

$\begin{matrix}{{H_{1\; \varphi} = {{Ae}^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},} & (4) \\{{E_{1\; p} = {{A\left( \frac{- u_{1}}{\sigma_{1} + {j\; \omega \; ɛ_{1}}} \right)}e^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},{and}} & (5) \\{E_{1z} = {{A\left( \frac{{- j}\; \gamma}{\sigma_{1} + {j\; \omega \; ɛ_{1}}} \right)}e^{u_{1}z}{{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}.}}} & (6)\end{matrix}$

In these expressions, z is the vertical coordinate normal to the surfaceof Region 1 and ρ is the radial coordinate, H_(n) ⁽²⁾(−jγρ) is a complexargument Hankel function of the second kind and order n, u₁ is thepropagation constant in the positive vertical (z) direction in Region 1,u₂ is the propagation constant in the vertical (z) direction in Region2, σ₁ is the conductivity of Region 1, ω is equal to 2πf, where f is afrequency of excitation, ε_(O) is the permittivity of free space, ε₁ isthe permittivity of Region 1, A is a source constant imposed by thesource, and γ is a surface wave radial propagation constant.

The propagation constants in the ±z directions are determined byseparating the wave equation above and below the interface betweenRegions 1 and 2, and imposing the boundary conditions. This exercisegives, in Region 2,

$\begin{matrix}{u_{2} = \frac{- {jk}_{o}}{\sqrt{1 + \left( {ɛ_{r} - {jx}} \right)}}} & (7)\end{matrix}$

and gives, in Region 1,

u ₁ =−u ₂(ε_(r) −jx).  (8)

The radial propagation constant γ is given by

$\begin{matrix}{{\gamma = {{j\sqrt{k_{o}^{2} + u_{2}^{2}}} = {j\frac{k_{o}n}{\sqrt{1 + n^{2}}}}}},} & (9)\end{matrix}$

which is a complex expression where n is the complex index of refractiongiven by

n=√{square root over (ε_(r) −jx)}.  (10)

In all of the above Equations,

$\begin{matrix}{{x = \frac{\sigma_{1}}{\omega \; ɛ_{o}}},{and}} & (11) \\{{k_{o} = {{\omega \sqrt{\mu_{o}ɛ_{o}}} = \frac{\lambda_{o}}{2\; \pi}}},} & (12)\end{matrix}$

where ε_(r) comprises the relative permittivity of Region 1, σ₁ is theconductivity of Region 1, ε_(o) is the permittivity of free space, andμ_(o) comprises the permeability of free space. Thus, the generatedsurface wave propagates parallel to the interface and exponentiallydecays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be considered to be acylindrically-symmetric, radially-propagating waveguide mode. SeeBarlow, H. M., and Brown, J., Radio Surface Waves, Oxford UniversityPress, 1962, pp. 10-12, 29-33. The present disclosure details structuresthat excite this “open boundary” waveguide mode. Specifically, accordingto various embodiments, a guided surface waveguide probe is providedwith a charge terminal of appropriate size that is fed with voltageand/or current and is positioned relative to the boundary interfacebetween Region 2 and Region 1. This may be better understood withreference to FIG. 3, which shows an example of a guided surfacewaveguide probe 300 a that includes a charge terminal T₁ elevated abovea lossy conducting medium 303 (e.g., the earth) along a vertical axis zthat is normal to a plane presented by the lossy conducting medium 303.The lossy conducting medium 303 makes up Region 1, and a second medium306 makes up Region 2 and shares a boundary interface with the lossyconducting medium 303.

According to one embodiment, the lossy conducting medium 303 cancomprise a terrestrial medium such as the planet Earth. To this end,such a terrestrial medium comprises all structures or formationsincluded thereon whether natural or man-made. For example, such aterrestrial medium can comprise natural elements such as rock, soil,sand, fresh water, sea water, trees, vegetation, and all other naturalelements that make up our planet. In addition, such a terrestrial mediumcan comprise man-made elements such as concrete, asphalt, buildingmaterials, and other man-made materials. In other embodiments, the lossyconducting medium 303 can comprise some medium other than the Earth,whether naturally occurring or man-made. In other embodiments, the lossyconducting medium 303 can comprise other media such as man-made surfacesand structures such as automobiles, aircraft, man-made materials (suchas plywood, plastic sheeting, or other materials) or other media.

In the case where the lossy conducting medium 303 comprises aterrestrial medium or Earth, the second medium 306 can comprise theatmosphere above the ground. As such, the atmosphere can be termed an“atmospheric medium” that comprises air and other elements that make upthe atmosphere of the Earth. In addition, it is possible that the secondmedium 306 can comprise other media relative to the lossy conductingmedium 303.

The guided surface waveguide probe 300 a includes a feed network 309that couples an excitation source 312 to the charge terminal T₁ via,e.g., a vertical feed line conductor. According to various embodiments,a charge Q₁ is imposed on the charge terminal T₁ to synthesize anelectric field based upon the voltage applied to terminal T₁ at anygiven instant. Depending on the angle of incidence (θ_(i)) of theelectric field (E), it is possible to substantially mode-match theelectric field to a guided surface waveguide mode on the surface of thelossy conducting medium 303 comprising Region 1.

By considering the Zenneck closed-form solutions of Equations (1)-(6),the Leontovich impedance boundary condition between Region 1 and Region2 can be stated as

{circumflex over (z)}×{right arrow over (H)} ₂(ρ,φ,0)={right arrow over(J)} _(S),  (13)

where {circumflex over (z)} is a unit normal in the positive vertical(+z) direction and {right arrow over (H)}₂ is the magnetic fieldstrength in Region 2 expressed by Equation (1) above. Equation (13)implies that the electric and magnetic fields specified in Equations(1)-(3) may result in a radial surface current density along theboundary interface, where the radial surface current density can bespecified by

J _(ρ)(ρ′)−AH ₁ ⁽²⁾(−jγρ′)  (14)

where A is a constant. Further, it should be noted that close-in to theguided surface waveguide probe 300 (for ρ<<λ), Equation (14) above hasthe behavior

$\begin{matrix}{{J_{close}\left( \rho^{\prime} \right)} = {\frac{- {A\left( {j\; 2} \right)}}{\pi \left( {{- j}\; \gamma \; \rho^{\prime}} \right)} = {{- H_{\varphi}} = {- {\frac{I_{o}}{2\; \pi \; \rho^{\prime}}.}}}}} & (15)\end{matrix}$

The negative sign means that when source current (I_(o)) flowsvertically upward as illustrated in FIG. 3, the “close-in” groundcurrent flows radially inward. By field matching on H_(ϕ) “close-in,” itcan be determined that

$\begin{matrix}{A = {{- \frac{I_{o}\gamma}{4}} = {- \frac{\omega \; q_{1}\gamma}{4}}}} & (16)\end{matrix}$

where q₁=C₁V₁, in Equations (1)-(6) and (14). Therefore, the radialsurface current density of Equation (14) can be restated as

$\begin{matrix}{{J_{\rho}\left( \rho^{\prime} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho^{\prime}} \right)}.}}} & (17)\end{matrix}$

The fields expressed by Equations (1)-(6) and (17) have the nature of atransmission line mode bound to a lossy interface, not radiation fieldsthat are associated with groundwave propagation. See Barlow, H. M. andBrown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5.

At this point, a review of the nature of the Hankel functions used inEquations (1)-(6) and (17) is provided for these solutions of the waveequation. One might observe that the Hankel functions of the first andsecond kind and order n are defined as complex combinations of thestandard Bessel functions of the first and second kinds

H _(n) ⁽¹⁾(x)=J _(n)(x)+jN _(n)(x), and  (18)

H _(n) ⁽²⁾(x)=J _(n)(x)−jN _(n)(x),  (19)

These functions represent cylindrical waves propagating radially inward(H⁽¹⁾) and outward (H⁽²⁾), respectively. The definition is analogous tothe relationship e^(±jx)=cos x±j sin x. See, for example, Harrington, R.F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460-463.

That (H⁽²⁾)(k_(ρ)ρ) is an outgoing wave can be recognized from its largeargument asymptotic behavior that is obtained directly from the seriesdefinitions of J_(n)(x) and N_(n)(x). Far-out from the guided surfacewaveguide probe:

$\begin{matrix}{{{{{H_{n}^{(2)}(x)}\underset{x\rightarrow\infty}{}\sqrt{\frac{2j}{\pi \; x}}}j^{n}e^{- {jx}}} = {\sqrt{\frac{2}{\pi x}}j^{n}e^{- {j{({x - \frac{\pi}{4}})}}}}},} & \left( {20a} \right)\end{matrix}$

which, when multiplied by e^(jωt), is an outward propagating cylindricalwave of the form e^(j(ωt-kρ)) with a 1/√{square root over (ρ)} spatialvariation. The first order (n=1) solution can be determined fromEquation (20a) to be

$\begin{matrix}{{{{H_{1}^{(2)}(x)}\underset{x\rightarrow\infty}{}j}\sqrt{\frac{2j}{\pi \; x}}e^{- {jx}}} = {\sqrt{\frac{2}{\pi \; x}}{e^{- {j{({x - \frac{\pi}{2} - \frac{\pi}{4}})}}}.}}} & \left( {20b} \right)\end{matrix}$

Close-in to the guided surface waveguide probe (for ρ<<λ), the Hankelfunction of first order and the second kind behaves as

$\begin{matrix}{{{H_{1}^{(2)}(x)}\underset{x\rightarrow 0}{}\frac{2j}{\pi \; x}}.} & (21)\end{matrix}$

Note that these asymptotic expressions are complex quantities. When x isa real quantity, Equations (20b) and (21) differ in phase by √{squareroot over (j)}, which corresponds to an extra phase advance or “phaseboost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotesof the first order Hankel function of the second kind have a Hankel“crossover” or transition point where they are of equal magnitude at adistance of ρ=R_(x).

Thus, beyond the Hankel crossover point the “far out” representationpredominates over the “close-in” representation of the Hankel function.The distance to the Hankel crossover point (or Hankel crossoverdistance) can be found by equating Equations (20b) and (21) for −jγρ,and solving for R_(x). With x=σ/ωε_(o), it can be seen that the far-outand close-in Hankel function asymptotes are frequency dependent, withthe Hankel crossover point moving out as the frequency is lowered. Itshould also be noted that the Hankel function asymptotes may also varyas the conductivity (σ) of the lossy conducting medium changes. Forexample, the conductivity of the soil can vary with changes in weatherconditions.

Referring to FIG. 4, shown is an example of a plot of the magnitudes ofthe first order Hankel functions of Equations (20b) and (21) for aRegion 1 conductivity of σ=0.010 mhos/m and relative permittivityε_(r)=15, at an operating frequency of 1850 kHz. Curve 403 is themagnitude of the far-out asymptote of Equation (20b) and curve 406 isthe magnitude of the close-in asymptote of Equation (21), with theHankel crossover point 409 occurring at a distance of R_(x)=54 feet.While the magnitudes are equal, a phase offset exists between the twoasymptotes at the Hankel crossover point 409. It can also be seen thatthe Hankel crossover distance is much less than a wavelength of theoperation frequency.

Considering the electric field components given by Equations (2) and (3)of the Zenneck closed-form solution in Region 2, it can be seen that theratio of E_(Z) and E_(ρ) asymptotically passes to

$\begin{matrix}{\frac{E_{z}}{E_{\rho}} = {{\left( \frac{{- j}\; \gamma}{u_{2}} \right){\frac{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}\underset{\rho\rightarrow\infty}{}\sqrt{ɛ_{r} - {j\frac{\sigma}{\omega \; ɛ_{o}}}}}} = {n = {\tan \; {\theta_{i}.}}}}} & (22)\end{matrix}$

where n is the complex index of refraction of Equation (10) and θ_(i) isthe angle of incidence of the electric field. In addition, the verticalcomponent of the mode-matched electric field of Equation (3)asymptotically passes to

$\begin{matrix}{{{E_{2z}\underset{\rho\rightarrow\infty}{}\left( \frac{q_{free}}{ɛ_{o}} \right)}\sqrt{\frac{\gamma^{3}}{8\; \pi}}e^{{- u_{2}}z}\frac{e^{- {j{({{\gamma \; \rho} - {\pi/4}})}}}}{\sqrt{\rho}}},} & (23)\end{matrix}$

which is linearly proportional to free charge on the isolated componentof the elevated charge terminal's capacitance at the terminal voltage,q_(free)=C_(free)×V_(T).

For example, the height H₁ of the elevated charge terminal T₁ in FIG. 3affects the amount of free charge on the charge terminal T₁. When thecharge terminal T₁ is near the ground plane of Region 1, most of thecharge Q₁ on the terminal is “bound.” As the charge terminal T₁ iselevated, the bound charge is lessened until the charge terminal T₁reaches a height at which substantially all of the isolated charge isfree.

The advantage of an increased capacitive elevation for the chargeterminal T₁ is that the charge on the elevated charge terminal T₁ isfurther removed from the ground plane, resulting in an increased amountof free charge q_(free) to couple energy into the guided surfacewaveguide mode. As the charge terminal T₁ is moved away from the groundplane, the charge distribution becomes more uniformly distributed aboutthe surface of the terminal. The amount of free charge is related to theself-capacitance of the charge terminal T₁.

For example, the capacitance of a spherical terminal can be expressed asa function of physical height above the ground plane. The capacitance ofa sphere at a physical height of h above a perfect ground is given by

C _(elevated sphere)=4πε_(o) a(1M+M ² +M ³+2M ⁴3M ⁵− . . . ),  (24)

where the diameter of the sphere is 2a, and where M=a/2h with h beingthe height of the spherical terminal. As can be seen, an increase in theterminal height h reduces the capacitance C of the charge terminal. Itcan be shown that for elevations of the charge terminal T₁ that are at aheight of about four times the diameter (4D=8a) or greater, the chargedistribution is approximately uniform about the spherical terminal,which can improve the coupling into the guided surface waveguide mode.

In the case of a sufficiently isolated terminal, the self-capacitance ofa conductive sphere can be approximated by C=4πε_(o)a, where a is theradius of the sphere in meters, and the self-capacitance of a disk canbe approximated by C=8ε_(o)a, where a is the radius of the disk inmeters. The charge terminal T₁ can include any shape such as a sphere, adisk, a cylinder, a cone, a torus, a hood, one or more rings, or anyother randomized shape or combination of shapes. An equivalent sphericaldiameter can be determined and used for positioning of the chargeterminal T₁.

This may be further understood with reference to the example of FIG. 3,where the charge terminal T₁ is elevated at a physical height ofh_(p)=H₁ above the lossy conducting medium 303. To reduce the effects ofthe “bound” charge, the charge terminal T₁ can be positioned at aphysical height that is at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ to reduce thebounded charge effects.

Referring next to FIG. 5A, shown is a ray optics interpretation of theelectric field produced by the elevated charge Q₁ on charge terminal T₁of FIG. 3. As in optics, minimizing the reflection of the incidentelectric field can improve and/or maximize the energy coupled into theguided surface waveguide mode of the lossy conducting medium 303. For anelectric field (E_(∥)) that is polarized parallel to the plane ofincidence (not the boundary interface), the amount of reflection of theincident electric field may be determined using the Fresnel reflectioncoefficient, which can be expressed as

$\begin{matrix}{{{\Gamma_{\parallel}\left( \theta_{i} \right)} = {\frac{E_{\parallel {,R}}}{E_{\parallel {,i}}} = \frac{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} - {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} + {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}}}},} & (25)\end{matrix}$

where θ_(i) is the conventional angle of incidence measured with respectto the surface normal.

In the example of FIG. 5A, the ray optic interpretation shows theincident field polarized parallel to the plane of incidence having anangle of incidence of θ_(i), which is measured with respect to thesurface normal ({circumflex over (z)}). There will be no reflection ofthe incident electric field when Γ_(∥)(θ_(i))=0 and thus the incidentelectric field will be completely coupled into a guided surfacewaveguide mode along the surface of the lossy conducting medium 303. Itcan be seen that the numerator of Equation (25) goes to zero when theangle of incidence is

θ_(i)=arctan(√{square root over (ε_(r) −jx)})=θ_(i,B),  (26)

where x=σ/ωε_(o). This complex angle of incidence (θ_(i,B)) is referredto as the Brewster angle. Referring back to Equation (22), it can beseen that the same complex Brewster angle (θ_(i,B)) relationship ispresent in both Equations (22) and (26).

As illustrated in FIG. 5A, the electric field vector E can be depictedas an incoming non-uniform plane wave, polarized parallel to the planeof incidence. The electric field vector E can be created fromindependent horizontal and vertical components as

{right arrow over (E)}(θ_(i))=E _(ρ) {circumflex over (ρ)}+E _(Z){circumflex over (z)}.  (27)

Geometrically, the illustration in FIG. 5A suggests that the electricfield vector E can be given by

$\begin{matrix}{{{E_{\rho}\left( {\rho,z} \right)} = {{E\left( {\rho,z} \right)}\cos \; \theta_{i}}},{and}} & \left( {28a} \right) \\{{{E_{z}\left( {\rho,z} \right)} = {{{E\left( {\rho,z} \right)}{\cos \left( {\frac{\pi}{2} - \theta_{i}} \right)}} = {{E\left( {\rho,z} \right)}\sin \; \theta_{i}}}},} & \left( {28b} \right)\end{matrix}$

which means that the field ratio is

$\begin{matrix}{\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \; \theta_{i}} = {\tan \; {\psi_{i}.}}}} & (29)\end{matrix}$

A generalized parameter W, called “wave tilt,” is noted herein as theratio of the horizontal electric field component to the verticalelectric field component given by

$\begin{matrix}{{W = {\frac{E_{\rho}}{E_{z}} = {{W}e^{j\; \Psi}}}},{or}} & \left( {30a} \right) \\{{\frac{1}{W} = {\frac{E_{z}}{E_{\rho}} = {{\tan \; \theta_{i}} = {\frac{1}{W}e^{{- j}\; \Psi}}}}},} & \left( {30b} \right)\end{matrix}$

which is complex and has both magnitude and phase. For anelectromagnetic wave in Region 2, the wave tilt angle (Ψ) is equal tothe angle between the normal of the wave-front at the boundary interfacewith Region 1 and the tangent to the boundary interface. This may beeasier to see in FIG. 5B, which illustrates equi-phase surfaces of anelectromagnetic wave and their normals for a radial cylindrical guidedsurface wave. At the boundary interface (z=0) with a perfect conductor,the wave-front normal is parallel to the tangent of the boundaryinterface, resulting in W=0. However, in the case of a lossy dielectric,a wave tilt W exists because the wave-front normal is not parallel withthe tangent of the boundary interface at z=0.

Applying Equation (30b) to a guided surface wave gives

$\begin{matrix}{{\tan \; \theta_{i,B}} = {\frac{E_{z}}{E_{\rho}} = {\frac{u_{2}}{\gamma} = {\sqrt{ɛ_{r} - {jx}} = {n = {\frac{1}{W} = {\frac{1}{W}{e^{{- j}\; \Psi}.}}}}}}}} & (31)\end{matrix}$

With the angle of incidence equal to the complex Brewster angle(θ_(i,B)), the Fresnel reflection coefficient of Equation (25) vanishes,as shown by

$\begin{matrix}{{{{\Gamma_{\parallel}\left( \theta_{i,B} \right)} = \frac{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} - {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} + {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}}}}_{\theta_{i} = \theta_{i,B}} = 0.} & (32)\end{matrix}$

By adjusting the complex field ratio of Equation (22), an incident fieldcan be synthesized to be incident at a complex angle at which thereflection is reduced or eliminated. Establishing this ratio asn=√{square root over (ε_(r)−jx)} results in the synthesized electricfield being incident at the complex Brewster angle, making thereflections vanish.

The concept of an electrical effective height can provide furtherinsight into synthesizing an electric field with a complex angle ofincidence with a guided surface waveguide probe 300. The electricaleffective height (h_(eff)) has been defined as

$\begin{matrix}{h_{eff} = {\frac{1}{I_{0}}{\int_{0}^{h_{p}}{{I(z)}{dz}}}}} & (33)\end{matrix}$

for a monopole with a physical height (or length) of h_(p). Since theexpression depends upon the magnitude and phase of the sourcedistribution along the structure, the effective height (or length) iscomplex in general. The integration of the distributed current I(z) ofthe structure is performed over the physical height of the structure(h_(p)), and normalized to the ground current (I₀) flowing upwardthrough the base (or input) of the structure. The distributed currentalong the structure can be expressed by

I(z)=I _(C) cos(β₀ z),  (34)

where β₀ is the propagation factor for current propagating on thestructure. In the example of FIG. 3, I_(C) is the current that isdistributed along the vertical structure of the guided surface waveguideprobe 300 a.

For example, consider a feed network 309 that includes a low loss coil(e.g., a helical coil) at the bottom of the structure and a verticalfeed line conductor connected between the coil and the charge terminalT₁. The phase delay due to the coil (or helical delay line) isθ_(c)=β_(p)l_(C), with a physical length of l_(C) and a propagationfactor of

$\begin{matrix}{{\beta_{p} = {\frac{2\; \pi}{\lambda_{p}} = \frac{2\; \pi}{V_{f}\lambda_{0}}}},} & (35)\end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). The phase delay ismeasured relative to the ground (stake) current I₀.

In addition, the spatial phase delay along the length l_(w) of thevertical feed line conductor can be given by θ_(y)=β_(w)l_(w) whereβ_(w) is the propagation phase constant for the vertical feed lineconductor. In some implementations, the spatial phase delay may beapproximated by θ_(y)=β_(w)h_(p), since the difference between thephysical height h_(p) of the guided surface waveguide probe 300 a andthe vertical feed line conductor length l_(w) is much less than awavelength at the supplied frequency (λ₀). As a result, the total phasedelay through the coil and vertical feed line conductor isΦ=θ_(c)+θ_(y), and the current fed to the top of the coil from thebottom of the physical structure is

I _(C)(θ_(c)+θ_(y))=I ₀ e ^(jΦ),  (36)

with the total phase delay Φ measured relative to the ground (stake)current I₀. Consequently, the electrical effective height of a guidedsurface waveguide probe 300 can be approximated by

$\begin{matrix}{{h_{eff} = {{\frac{1}{I_{0}}{\int_{0}^{h_{p}}{I_{0}e^{j\; \Phi}{\cos \left( {\beta_{0}z} \right)}{dz}}}} \cong {h_{p}e^{j\; \Phi}}}},} & (37)\end{matrix}$

for the case where the physical height h_(p)<<λ₀. The complex effectiveheight of a monopole, h_(eff), =h_(p) at an angle (or phase shift) of Φ,may be adjusted to cause the source fields to match a guided surfacewaveguide mode and cause a guided surface wave to be launched on thelossy conducting medium 303.

In the example of FIG. 5A, ray optics are used to illustrate the complexangle trigonometry of the incident electric field (E) having a complexBrewster angle of incidence (θ_(i,B)) at the Hankel crossover distance(R_(x)) 315. Recall from Equation (26) that, for a lossy conductingmedium, the Brewster angle is complex and specified by

$\begin{matrix}{{\tan \; \theta_{i,B}} = {\sqrt{ɛ_{r} - {j\frac{\sigma}{\omega \; ɛ_{o}}}} = {n.}}} & (38)\end{matrix}$

Electrically, the geometric parameters are related by the electricaleffective height (h_(eff)) of the charge terminal T₁ by

R _(x) tan ψ_(i,B) =R _(x) ×W=h _(eff) =h _(p) e ^(jΦ),  (39)

where ψ_(i,B)=(π/2)−θ_(i,B) is the Brewster angle measured from thesurface of the lossy conducting medium. To couple into the guidedsurface waveguide mode, the wave tilt of the electric field at theHankel crossover distance can be expressed as the ratio of theelectrical effective height and the Hankel crossover distance

$\begin{matrix}{\frac{h_{eff}}{R_{x}} = {{\tan \; \psi_{i,B}} = {W_{Rx}.}}} & (40)\end{matrix}$

Since both the physical height (h_(p)) and the Hankel crossover distance(R_(x)) are real quantities, the angle (Ψ) of the desired guided surfacewave tilt at the Hankel crossover distance (R_(x)) is equal to the phase(Φ) of the complex effective height (h_(eff)). This implies that byvarying the phase at the supply point of the coil, and thus the phaseshift in Equation (37), the phase, Φ, of the complex effective heightcan be manipulated to match the angle of the wave tilt, Ψ, of the guidedsurface waveguide mode at the Hankel crossover point 315: Φ=Ψ.

In FIG. 5A, a right triangle is depicted having an adjacent side oflength R_(x) along the lossy conducting medium surface and a complexBrewster angle ψ_(i,B) measured between a ray 316 extending between theHankel crossover point 315 at R_(x) and the center of the chargeterminal T₁, and the lossy conducting medium surface 317 between theHankel crossover point 315 and the charge terminal T₁. With the chargeterminal T₁ positioned at physical height h_(p) and excited with acharge having the appropriate phase delay Φ, the resulting electricfield is incident with the lossy conducting medium boundary interface atthe Hankel crossover distance R_(x), and at the Brewster angle. Underthese conditions, the guided surface waveguide mode can be excitedwithout reflection or substantially negligible reflection.

If the physical height of the charge terminal T₁ is decreased withoutchanging the phase shift Φ of the effective height (h_(eff)), theresulting electric field intersects the lossy conducting medium 303 atthe Brewster angle at a reduced distance from the guided surfacewaveguide probe 300. FIG. 6 graphically illustrates the effect ofdecreasing the physical height of the charge terminal T₁ on the distancewhere the electric field is incident at the Brewster angle. As theheight is decreased from h₃ through h₂ to h₁, the point where theelectric field intersects with the lossy conducting medium (e.g., theearth) at the Brewster angle moves closer to the charge terminalposition. However, as Equation (39) indicates, the height H₁ (FIG. 3) ofthe charge terminal T₁ should be at or higher than the he physicalheight (h_(p)) in order to excite the far-out component of the Hankelfunction. With the charge terminal T₁ positioned at or above theeffective height (h_(eff)), the lossy conducting medium 303 can beilluminated at the Brewster angle of incidence (ψ_(i,B)=(π/2)−θ_(i,B))at or beyond the Hankel crossover distance (R_(x)) 315 as illustrated inFIG. 5A. To reduce or minimize the bound charge on the charge terminalT₁, the height should be at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ as mentionedabove.

A guided surface waveguide probe 300 can be configured to establish anelectric field having a wave tilt that corresponds to a waveilluminating the surface of the lossy conducting medium 303 at a complexBrewster angle, thereby exciting radial surface currents bysubstantially mode-matching to a guided surface wave mode at (or beyond)the Hankel crossover point 315 at R_(x). Referring to FIG. 7, shown is agraphical representation of an example of a guided surface waveguideprobe 300 b that includes a charge terminal T₁. An AC source 712 acts asthe excitation source (312 of FIG. 3) for the charge terminal T₁, whichis coupled to the guided surface waveguide probe 300 b through a feednetwork (309 of FIG. 3) comprising a coil 709 such as, e.g., a helicalcoil. In other implementations, the AC source 712 can be inductivelycoupled to the coil 709 through a primary coil. In some embodiments, animpedance matching network may be included to improve and/or maximizecoupling of the AC source 712 to the coil 709.

As shown in FIG. 7, the guided surface waveguide probe 300 b can includethe upper charge terminal T₁ (e.g., a sphere at height h_(p)) that ispositioned along a vertical axis z that is substantially normal to theplane presented by the lossy conducting medium 303. A second medium 306is located above the lossy conducting medium 303. The charge terminal T₁has a self-capacitance C_(T). During operation, charge Q₁ is imposed onthe terminal T₁ depending on the voltage applied to the terminal T₁ atany given instant.

In the example of FIG. 7, the coil 709 is coupled to a ground stake 715at a first end and to the charge terminal T₁ via a vertical feed lineconductor 718. In some implementations, the coil connection to thecharge terminal T₁ can be adjusted using a tap 721 of the coil 709 asshown in FIG. 7. The coil 709 can be energized at an operating frequencyby the AC source 712 through a tap 724 at a lower portion of the coil709. In other implementations, the AC source 712 can be inductivelycoupled to the coil 709 through a primary coil.

The construction and adjustment of the guided surface waveguide probe300 is based upon various operating conditions, such as the transmissionfrequency, conditions of the lossy conducting medium (e.g., soilconductivity σ and relative permittivity ε_(r)), and size of the chargeterminal T₁. The index of refraction can be calculated from Equations(10) and (11) as

n=√{square root over (ε_(r) −jx)},  (41)

where x=σ/ωε_(o) with ω=2πf. The conductivity σ and relativepermittivity ε_(r) can be determined through test measurements of thelossy conducting medium 303. The complex Brewster angle (θ_(i,B))measured from the surface normal can also be determined from Equation(26) as

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)}),  (42)

or measured from the surface as shown in FIG. 5A as

$\begin{matrix}{\psi_{i,B} = {\frac{\pi}{2} - {\theta_{i,B}.}}} & (43)\end{matrix}$

The wave tilt at the Hankel crossover distance (W_(Rx)) can also befound using Equation (40).

The Hankel crossover distance can also be found by equating themagnitudes of Equations (20b) and (21) for −jγρ, and solving for R_(x)as illustrated by FIG. 4. The electrical effective height can then bedetermined from Equation (39) using the Hankel crossover distance andthe complex Brewster angle as

h _(eff) =h _(p) e ^(jΦ) =R _(x) tan ψ_(i,B).  (44)

As can be seen from Equation (44), the complex effective height(h_(eff)) includes a magnitude that is associated with the physicalheight (h_(p)) of the charge terminal T₁ and a phase delay (Φ) that isto be associated with the angle (Ψ) of the wave tilt at the Hankelcrossover distance (R_(x)). With these variables and the selected chargeterminal T₁ configuration, it is possible to determine the configurationof a guided surface waveguide probe 300.

With the charge terminal T₁ positioned at or above the physical height(h_(p)), the feed network (309 of FIG. 3) and/or the vertical feed lineconnecting the feed network to the charge terminal T₁ can be adjusted tomatch the phase (Φ) of the charge Q₁ on the charge terminal T₁ to theangle (Ψ) of the wave tilt (W). The size of the charge terminal T₁ canbe chosen to provide a sufficiently large surface for the charge Q₁imposed on the terminals. In general, it is desirable to make the chargeterminal T₁ as large as practical. The size of the charge terminal T₁should be large enough to avoid ionization of the surrounding air, whichcan result in electrical discharge or sparking around the chargeterminal.

The phase delay θ_(c) of a helically-wound coil can be determined fromMaxwell's equations as has been discussed by Corum, K. L. and J. F.Corum, “RF Coils, Helical Resonators and Voltage Magnification byCoherent Spatial Modes,” Microwave Review, Vol. 7, No. 2, September2001, pp. 36-45., which is incorporated herein by reference in itsentirety. For a helical coil with H/D>1, the ratio of the velocity ofpropagation (v) of a wave along the coil's longitudinal axis to thespeed of light (c), or the “velocity factor,” is given by

$\begin{matrix}{{V_{f} = {\frac{\upsilon}{c} = \frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}}}},} & (45)\end{matrix}$

where H is the axial length of the solenoidal helix, D is the coildiameter, N is the number of turns of the coil, s=H/N is theturn-to-turn spacing (or helix pitch) of the coil, and λ_(o) is thefree-space wavelength. Based upon this relationship, the electricallength, or phase delay, of the helical coil is given by

$\begin{matrix}{\theta_{c} = {{\beta_{p}H} = {{\frac{2\; \pi}{\lambda_{p}}H} = {\frac{2\; \pi}{V_{f}\lambda_{0}}{H.}}}}} & (46)\end{matrix}$

The principle is the same if the helix is wound spirally or is short andfat, but V_(f) and θ_(c) are easier to obtain by experimentalmeasurement. The expression for the characteristic (wave) impedance of ahelical transmission line has also been derived as

$\begin{matrix}{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\; {n\left( \frac{V_{f}\lambda_{0}}{D} \right)}} - 1.027} \right\rbrack}.}} & (47)\end{matrix}$

The spatial phase delay θ_(y) of the structure can be determined usingthe traveling wave phase delay of the vertical feed line conductor 718(FIG. 7). The capacitance of a cylindrical vertical conductor above aprefect ground plane can be expressed as

$\begin{matrix}{{C_{A} = {\frac{2\; \pi \; ɛ_{o}h_{w}}{{\; {n\left( \frac{h}{a} \right)}} - 1}{Farads}}},} & (48)\end{matrix}$

where h_(w) is the vertical length (or height) of the conductor and a isthe radius (in mks units). As with the helical coil, the traveling wavephase delay of the vertical feed line conductor can be given by

$\begin{matrix}{{\theta_{y} = {{\beta_{w}h_{w}} = {{\frac{2\; \pi}{\lambda_{w}}h_{w}} = {\frac{2\; \pi}{V_{w}\lambda_{0}}h_{w}}}}},} & (49)\end{matrix}$

where β_(w) is the propagation phase constant for the vertical feed lineconductor, h_(w) is the vertical length (or height) of the vertical feedline conductor, V_(w) is the velocity factor on the wire, λ₀ is thewavelength at the supplied frequency, and λ_(w) is the propagationwavelength resulting from the velocity factor V_(w). For a uniformcylindrical conductor, the velocity factor is a constant withV_(w)≈0.94, or in a range from about 0.93 to about 0.98. If the mast isconsidered to be a uniform transmission line, its average characteristicimpedance can be approximated by

$\begin{matrix}{{Z_{w} = {\frac{60}{V_{w}}\left\lbrack {{{n}\left( \frac{_{w}}{} \right)} - 1} \right\rbrack}},} & (50)\end{matrix}$

where V_(w)≈0.94 for a uniform cylindrical conductor and a is the radiusof the conductor. An alternative expression that has been employed inamateur radio literature for the characteristic impedance of asingle-wire feed line can be given by

$\begin{matrix}{Z_{w} = {138\; {{\log \left( \frac{1.123\; V_{w}\lambda_{0}}{2\; \pi \; a} \right)}.}}} & (51)\end{matrix}$

Equation (51) implies that Z_(o) for a single-wire feeder varies withfrequency. The phase delay can be determined based upon the capacitanceand characteristic impedance.

With a charge terminal T₁ positioned over the lossy conducting medium303 as shown in FIG. 3, the feed network 309 can be adjusted to excitethe charge terminal T₁ with the phase shift (Φ) of the complex effectiveheight (h_(eff)) equal to the angle (Ψ) of the wave tilt at the Hankelcrossover distance, or Φ=Ψ. When this condition is met, the electricfield produced by the charge oscillating Q₁ on the charge terminal T₁ iscoupled into a guided surface waveguide mode traveling along the surfaceof a lossy conducting medium 303. For example, if the Brewster angle(θ_(i,B)), the phase delay (θ_(y)) associated with the vertical feedline conductor 718 (FIG. 7), and the configuration of the coil 709 (FIG.7) are known, then the position of the tap 721 (FIG. 7) can bedetermined and adjusted to impose an oscillating charge Q₁ on the chargeterminal T₁ with phase Φ=T. The position of the tap 721 may be adjustedto maximize coupling the traveling surface waves into the guided surfacewaveguide mode. Excess coil length beyond the position of the tap 721can be removed to reduce the capacitive effects. The vertical wireheight and/or the geometrical parameters of the helical coil may also bevaried.

The coupling to the guided surface waveguide mode on the surface of thelossy conducting medium 303 can be improved and/or optimized by tuningthe guided surface waveguide probe 300 for standing wave resonance withrespect to a complex image plane associated with the charge Q₁ on thecharge terminal T₁. By doing this, the performance of the guided surfacewaveguide probe 300 can be adjusted for increased and/or maximum voltage(and thus charge Q₁) on the charge terminal T₁. Referring back to FIG.3, the effect of the lossy conducting medium 303 in Region 1 can beexamined using image theory analysis.

Physically, an elevated charge Q₁ placed over a perfectly conductingplane attracts the free charge on the perfectly conducting plane, whichthen “piles up” in the region under the elevated charge Q₁. Theresulting distribution of “bound” electricity on the perfectlyconducting plane is similar to a bell-shaped curve. The superposition ofthe potential of the elevated charge Q₁, plus the potential of theinduced “piled up” charge beneath it, forces a zero equipotentialsurface for the perfectly conducting plane. The boundary value problemsolution that describes the fields in the region above the perfectlyconducting plane may be obtained using the classical notion of imagecharges, where the field from the elevated charge is superimposed withthe field from a corresponding “image” charge below the perfectlyconducting plane.

This analysis may also be used with respect to a lossy conducting medium303 by assuming the presence of an effective image charge Q₁′ beneaththe guided surface waveguide probe 300. The effective image charge Q₁′coincides with the charge Q₁ on the charge terminal T₁ about aconducting image ground plane 318, as illustrated in FIG. 3. However,the image charge Q₁′ is not merely located at some real depth and 180°out of phase with the primary source charge Q₁ on the charge terminalT₁, as they would be in the case of a perfect conductor. Rather, thelossy conducting medium 303 (e.g., a terrestrial medium) presents aphase shifted image. That is to say, the image charge Q₁′ is at acomplex depth below the surface (or physical boundary) of the lossyconducting medium 303. For a discussion of complex image depth,reference is made to Wait, J. R., “Complex Image Theory—Revisited,” IEEEAntennas and Propagation Magazine, Vol. 33, No. 4, August 1991, pp.27-29, which is incorporated herein by reference in its entirety.

Instead of the image charge Q₁′ being at a depth that is equal to thephysical height (H₁) of the charge Q₁, the conducting image ground plane318 (representing a perfect conductor) is located at a complex depth ofz=−d/2 and the image charge Q₁′ appears at a complex depth (i.e., the“depth” has both magnitude and phase), given by −D₁=−(d/2+d/2+H₁)≠H₁.For vertically polarized sources over the earth,

$\begin{matrix}{{d = {{\frac{2\sqrt{\gamma_{e}^{2} + k_{0}^{2}}}{\gamma_{e}^{2}} \approx \frac{2}{\gamma_{e}}} = {{d_{r} + {jd}_{i}} = {{d}{\angle\zeta}}}}},} & (52)\end{matrix}$

where

γ_(e) ² =jωμ ₁σ₁−ω²μ₁ε₁, and  (53)

k _(o)=ω√{square root over (μ_(o)ε_(o))}.  (54)

as indicated in Equation (12). The complex spacing of the image charge,in turn, implies that the external field will experience extra phaseshifts not encountered when the interface is either a dielectric or aperfect conductor. In the lossy conducting medium, the wave front normalis parallel to the tangent of the conducting image ground plane 318 atz=−d/2, and not at the boundary interface between Regions 1 and 2.

Consider the case illustrated in FIG. 8A where the lossy conductingmedium 303 is a finitely conducting earth 803 with a physical boundary806. The finitely conducting earth 803 may be replaced by a perfectlyconducting image ground plane 809 as shown in FIG. 8B, which is locatedat a complex depth z₁ below the physical boundary 806. This equivalentrepresentation exhibits the same impedance when looking down into theinterface at the physical boundary 806. The equivalent representation ofFIG. 8B can be modeled as an equivalent transmission line, as shown inFIG. 8C. The cross-section of the equivalent structure is represented asa (z-directed) end-loaded transmission line, with the impedance of theperfectly conducting image plane being a short circuit (z_(s)=0). Thedepth z₁ can be determined by equating the TEM wave impedance lookingdown at the earth to an image ground plane impedance z_(in) seen lookinginto the transmission line of FIG. 8C.

In the case of FIG. 8A, the propagation constant and wave intrinsicimpedance in the upper region (air) 812 are

$\begin{matrix}{{\gamma_{o} = {{j\; \omega \sqrt{\mu_{o}ɛ_{o}}} = {0 + {j\; \beta_{o}}}}},{and}} & (55) \\{Z_{o} = {\frac{j\; \omega \; \mu_{o}}{\gamma_{o}} = {\sqrt{\frac{\mu_{o}}{ɛ_{o}}}.}}} & (56)\end{matrix}$

In the lossy earth 803, the propagation constant and wave intrinsicimpedance are

$\begin{matrix}{{\gamma_{e} = \sqrt{j\; \omega \; {\mu_{1}\left( {\sigma_{1} + {j\; \omega \; ɛ_{1}}} \right)}}},{and}} & (57) \\{Z_{e} = {\frac{j\; \omega \; \mu_{1}}{\gamma_{e}}.}} & (58)\end{matrix}$

For normal incidence, the equivalent representation of FIG. 8B isequivalent to a TEM transmission line whose characteristic impedance isthat of air (z_(o)), with propagation constant of γ_(o), and whoselength is z₁. As such, the image ground plane impedance Z_(in) seen atthe interface for the shorted transmission line of FIG. 8C is given by

Z _(in) =Z _(o) tan h(γ_(o) z ₁).  (59)

Equating the image ground plane impedance Z_(in) associated with theequivalent model of FIG. 8C to the normal incidence wave impedance ofFIG. 8A and solving for z₁ gives the distance to a short circuit (theperfectly conducting image ground plane 809) as

$\begin{matrix}{{Z_{1} = {{\frac{1}{\gamma_{o}}\tan \; {h^{- 1}\left( \frac{Z_{e}}{Z_{o}} \right)}} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{\gamma_{o}}{\gamma_{e}} \right)}} \approx \frac{1}{\gamma_{e}}}}},} & (60)\end{matrix}$

where only the first term of the series expansion for the inversehyperbolic tangent is considered for this approximation. Note that inthe air region 812, the propagation constant is γ_(o)=jβ_(o), soZ_(in)=jZ_(o) tan β_(o)z₁ (which is a purely imaginary quantity for areal z₁), but z_(e) is a complex value if σ≠0. Therefore, Z_(in)=Z_(e)only when z₁ is a complex distance.

Since the equivalent representation of FIG. 8B includes a perfectlyconducting image ground plane 809, the image depth for a charge orcurrent lying at the surface of the earth (physical boundary 806) isequal to distance z₁ on the other side of the image ground plane 809, ord=2×z₁ beneath the earth's surface (which is located at z=0). Thus, thedistance to the perfectly conducting image ground plane 809 can beapproximated by

$\begin{matrix}{d = {{2\; z_{1}} \approx {\frac{2}{\gamma_{e}}.}}} & (61)\end{matrix}$

Additionally, the “image charge” will be “equal and opposite” to thereal charge, so the potential of the perfectly conducting image groundplane 809 at depth z₁=−d/2 will be zero.

If a charge Q₁ is elevated a distance H₁ above the surface of the earthas illustrated in FIG. 3, then the image charge Q₁′ resides at a complexdistance of D₁=d+H₁ below the surface, or a complex distance of d/2+H₁below the image ground plane 318. The guided surface waveguide probe 300b of FIG. 7 can be modeled as an equivalent single-wire transmissionline image plane model that can be based upon the perfectly conductingimage ground plane 809 of FIG. 8B. FIG. 9A shows an example of theequivalent single-wire transmission line image plane model and FIG. 9Billustrates an example of the equivalent classic transmission linemodel, including the shorted transmission line of FIG. 8C.

In the equivalent image plane models of FIGS. 9A and 9B, Φ=θ_(y)+θ_(c)is the traveling wave phase delay of the guided surface waveguide probe300 referenced to earth (or the lossy conducting medium 303),θ_(c)=β_(p)H is the electrical length of the coil 709 (FIG. 7), ofphysical length H, expressed in degrees, θ_(y)=β_(w)h_(w) is theelectrical length of the vertical feed line conductor 718 (FIG. 7), ofphysical length h_(w), expressed in degrees, and θ_(d)=β_(o)d/2 is thephase shift between the image ground plane 809 and the physical boundary806 of the earth (or lossy conducting medium 303). In the example ofFIGS. 9A and 9B, Z_(w) is the characteristic impedance of the elevatedvertical feed line conductor 718 in ohms, Z_(c) is the characteristicimpedance of the coil 709 in ohms, and Z_(O) is the characteristicimpedance of free space.

At the base of the guided surface waveguide probe 300, the impedanceseen “looking up” into the structure is Z_(↑)=Z_(base). With a loadimpedance of:

$\begin{matrix}{{Z_{L} = \frac{1}{j\; \omega \; C_{T}}},} & (62)\end{matrix}$

where C_(T) is the self-capacitance of the charge terminal T₁, theimpedance seen “looking up” into the vertical feed line conductor 718(FIG. 7) is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{w}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}{Z_{w} + {Z_{L}\tanh \; \left( {j\; \beta_{w}h_{w}} \right)}}} = {Z_{W}\frac{Z_{L} + {Z_{w}{\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{w} + {Z_{L}{\tanh \left( {j\; \theta_{y}} \right)}}}}}},} & (63)\end{matrix}$

and the impedance seen “looking up” into the coil 709 (FIG. 7) is givenby:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}{\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{c} + {Z_{2}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{c}{\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{c} + {Z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}.}}}} & (64)\end{matrix}$

At the base of the guided surface waveguide probe 300, the impedanceseen “looking down” into the lossy conducting medium 303 isZ_(↓)=Z_(in), which is given by:

$\begin{matrix}{{Z_{in} = {{Z_{o}\frac{Z_{s} + {Z_{o}{\tanh \left\lbrack {j\; {\beta_{o}\left( {d/2} \right)}} \right\rbrack}}}{Z_{o} + {Z_{s}{\tanh \left\lbrack {j\; {\beta_{o}\left( {d/2} \right)}} \right\rbrack}}}} = {Z_{o}{\tanh \left( {j\; \theta_{d}} \right)}}}},} & (65)\end{matrix}$

where Z_(s)=0.

Neglecting losses, the equivalent image plane model can be tuned toresonance when Z_(↓)+Z_(↑)=0 at the physical boundary 806. Or, in thelow loss case, X_(↓)+X_(↑)=0 at the physical boundary 806, where X isthe corresponding reactive component. Thus, the impedance at thephysical boundary 806 “looking up” into the guided surface waveguideprobe 300 is the conjugate of the impedance at the physical boundary 806“looking down” into the lossy conducting medium 303. By adjusting theload impedance Z_(L) of the charge terminal T₁ while maintaining thetraveling wave phase delay Φ equal to the angle of the media's wave tiltΨ, so that Φ=Ψ, which improves and/or maximizes coupling of the probe'selectric field to a guided surface waveguide mode along the surface ofthe lossy conducting medium 303 (e.g., earth), the equivalent imageplane models of FIGS. 9A and 9B can be tuned to resonance with respectto the image ground plane 809. In this way, the impedance of theequivalent complex image plane model is purely resistive, whichmaintains a superposed standing wave on the probe structure thatmaximizes the voltage and elevated charge on terminal T₁, and byequations (1)-(3) and (16) maximizes the propagating surface wave.

It follows from the Hankel solutions, that the guided surface waveexcited by the guided surface waveguide probe 300 is an outwardpropagating traveling wave. The source distribution along the feednetwork 309 between the charge terminal T₁ and the ground stake 715 ofthe guided surface waveguide probe 300 (FIGS. 3 and 7) is actuallycomposed of a superposition of a traveling wave plus a standing wave onthe structure. With the charge terminal T₁ positioned at or above thephysical height h_(p), the phase delay of the traveling wave movingthrough the feed network 309 is matched to the angle of the wave tiltassociated with the lossy conducting medium 303. This mode-matchingallows the traveling wave to be launched along the lossy conductingmedium 303. Once the phase delay has been established for the travelingwave, the load impedance Z_(L) of the charge terminal T₁ is adjusted tobring the probe structure into standing wave resonance with respect tothe image ground plane (318 of FIG. 3 or 809 of FIG. 8), which is at acomplex depth of −d/2. In that case, the impedance seen from the imageground plane has zero reactance and the charge on the charge terminal T₁is maximized.

The distinction between the traveling wave phenomenon and standing wavephenomena is that (1) the phase delay of traveling waves (θ=βd) on asection of transmission line of length d (sometimes called a “delayline”) is due to propagation time delays; whereas (2) theposition-dependent phase of standing waves (which are composed offorward and backward propagating waves) depends on both the line lengthpropagation time delay and impedance transitions at interfaces betweenline sections of different characteristic impedances. In addition to thephase delay that arises due to the physical length of a section oftransmission line operating in sinusoidal steady-state, there is anextra reflection coefficient phase at impedance discontinuities that isdue to the ratio of Z_(oa)/Z_(ob), where Z_(oa) and Z_(ob) are thecharacteristic impedances of two sections of a transmission line suchas, e.g., a helical coil section of characteristic impedanceZ_(oa)=Z_(c) (FIG. 9B) and a straight section of vertical feed lineconductor of characteristic impedance Z_(ob)=Z_(w) (FIG. 9B). The effectof such discontinuous phase jumps can be seen in the Smith chart plotsin FIG. 12.

As a result of this phenomenon, two relatively short transmission linesections of widely differing characteristic impedance may be used toprovide a very large phase shift. For example, a probe structurecomposed of two sections of transmission line, one of low impedance andone of high impedance, together totaling a physical length of, say,0.05λ, may be fabricated to provide a phase shift of 90° which isequivalent to a 0.25λ resonance. This is due to the large jump incharacteristic impedances. In this way, a physically short probestructure can be electrically longer than the two physical lengthscombined. This is illustrated in FIGS. 9A and 9B, but is especiallyclear in FIG. 12 where the discontinuities in the impedance ratiosprovide large jumps in phase between the different plotted sections onthe Smith chart. The impedance discontinuity provides a substantialphase shift where the sections are joined together.

Referring to FIG. 10, shown is a flow chart illustrating an example ofadjusting a guided surface waveguide probe 300 (FIG. 3) to substantiallymode-match to a guided surface waveguide mode on the surface of thelossy conducting medium, which launches a guided surface traveling wavealong the surface of a lossy conducting medium 303 (FIG. 3). Beginningwith 1003, the charge terminal T₁ of the guided surface waveguide probe300 is positioned at a defined height above a lossy conducting medium303. Utilizing the characteristics of the lossy conducting medium 303and the operating frequency of the guided surface waveguide probe 300,the Hankel crossover distance can also be found by equating themagnitudes of Equations (20b) and (21) for −jγρ, and solving for R_(x)as illustrated by FIG. 4. The complex index of refraction (n) can bedetermined using Equation (41) and the complex Brewster angle (θ_(i,B))can then be determined from Equation (42). The physical height (h_(p))of the charge terminal T₁ can then be determined from Equation (44). Thecharge terminal T₁ should be at or higher than the physical height(h_(p)) in order to excite the far-out component of the Hankel function.This height relationship is initially considered when launching surfacewaves. To reduce or minimize the bound charge on the charge terminal T₁,the height should be at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁.

At 1006, the electrical phase delay Φ of the elevated charge Q₁ on thecharge terminal T₁ is matched to the complex wave tilt angle Ψ. Thephase delay (θ_(c)) of the helical coil and/or the phase delay (θ_(y))of the vertical feed line conductor can be adjusted to make Φ equal tothe angle (Ψ) of the wave tilt (W). Based on Equation (31), the angle(Ψ) of the wave tilt can be determined from:

$\begin{matrix}{W = {\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \; \theta_{i,B}} = {\frac{1}{n} = {{W}{e^{j\; \Psi}.}}}}}} & (66)\end{matrix}$

The electrical phase Φ can then be matched to the angle of the wavetilt. This angular (or phase) relationship is next considered whenlaunching surface waves. For example, the electrical phase delayΦ=θ_(c)+θ_(y) can be adjusted by varying the geometrical parameters ofthe coil 709 (FIG. 7) and/or the length (or height) of the vertical feedline conductor 718 (FIG. 7). By matching Φ=Ψ, an electric field can beestablished at or beyond the Hankel crossover distance (R_(x)) with acomplex Brewster angle at the boundary interface to excite the surfacewaveguide mode and launch a traveling wave along the lossy conductingmedium 303.

Next at 1009, the load impedance of the charge terminal T₁ is tuned toresonate the equivalent image plane model of the guided surfacewaveguide probe 300. The depth (d/2) of the conducting image groundplane 809 (or 318 of FIG. 3) can be determined using Equations (52),(53) and (54) and the values of the lossy conducting medium 303 (e.g.,the earth), which can be measured. Using that depth, the phase shift(θ_(d)) between the image ground plane 809 and the physical boundary 806of the lossy conducting medium 303 can be determined using θ_(d)=β_(o)d/2. The impedance (Z_(in)) as seen “looking down” into the lossyconducting medium 303 can then be determined using Equation (65). Thisresonance relationship can be considered to maximize the launchedsurface waves.

Based upon the adjusted parameters of the coil 709 and the length of thevertical feed line conductor 718, the velocity factor, phase delay, andimpedance of the coil 709 and vertical feed line conductor 718 can bedetermined using Equations (45) through (51). In addition, theself-capacitance (C_(T)) of the charge terminal T₁ can be determinedusing, e.g., Equation (24). The propagation factor (β_(p)) of the coil709 can be determined using Equation (35) and the propagation phaseconstant (β_(w)) for the vertical feed line conductor 718 can bedetermined using Equation (49). Using the self-capacitance and thedetermined values of the coil 709 and vertical feed line conductor 718,the impedance (Z_(base)) of the guided surface waveguide probe 300 asseen “looking up” into the coil 709 can be determined using Equations(62), (63) and (64).

The equivalent image plane model of the guided surface waveguide probe300 can be tuned to resonance by adjusting the load impedance Z_(L) suchthat the reactance component X_(base) of Z_(base) cancels out thereactance component X_(in) of Z_(in), or X_(base)+X_(in)=0. Thus, theimpedance at the physical boundary 806 “looking up” into the guidedsurface waveguide probe 300 is the conjugate of the impedance at thephysical boundary 806 “looking down” into the lossy conducting medium303. The load impedance Z_(L) can be adjusted by varying the capacitance(C_(T)) of the charge terminal T₁ without changing the electrical phasedelay Φ=θ_(c)+θ_(y) of the charge terminal T₁. An iterative approach maybe taken to tune the load impedance Z_(L) for resonance of theequivalent image plane model with respect to the conducting image groundplane 809 (or 318). In this way, the coupling of the electric field to aguided surface waveguide mode along the surface of the lossy conductingmedium 303 (e.g., earth) can be improved and/or maximized.

This may be better understood by illustrating the situation with anumerical example. Consider a guided surface waveguide probe 300comprising a top-loaded vertical stub of physical height h_(p) with acharge terminal T₁ at the top, where the charge terminal T₁ is excitedthrough a helical coil and vertical feed line conductor at anoperational frequency (f_(o)) of 1.85 MHz. With a height (H₁) of 16 feetand the lossy conducting medium 303 (i.e., earth) having a relativepermittivity of ε_(r)=15 and a conductivity of σ₁=0.010 mhos/m, severalsurface wave propagation parameters can be calculated for f_(o)=1.850MHz. Under these conditions, the Hankel crossover distance can be foundto be R_(x)=54.5 feet with a physical height of h_(p)=5.5 feet, which iswell below the actual height of the charge terminal T₁. While a chargeterminal height of H₁=5.5 feet could have been used, the taller probestructure reduced the bound capacitance, permitting a greater percentageof free charge on the charge terminal T₁ providing greater fieldstrength and excitation of the traveling wave.

The wave length can be determined as:

$\begin{matrix}{{\lambda_{o} = {\frac{c}{f_{o}} = {162.162\mspace{14mu} {meters}}}},} & (67)\end{matrix}$

where c is the speed of light. The complex index of refraction is:

n=√{square root over (ε_(r) −jx)}=7.529−j6.546,  (68)

from Equation (41), where x=σ₁/ωε_(o) with ω=2πf_(o), and the complexBrewster angle is:

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)})=85.6−j3.744°.  (69)

from Equation (42). Using Equation (66), the wave tilt values can bedetermined to be:

$\begin{matrix}{W = {\frac{1}{\tan \; \theta_{i,B}} = {\frac{1}{n} = {{{W}e^{j\; \Psi}} = {0.101{e^{j\; 40.614{^\circ}}.}}}}}} & (70)\end{matrix}$

Thus, the helical coil can be adjusted to match Φ=Ψ=40.614°

The velocity factor of the vertical feed line conductor (approximated asa uniform cylindrical conductor with a diameter of 0.27 inches) can begiven as V_(w)≈0.93. Since h_(p)<<λ_(o), the propagation phase constantfor the vertical feed line conductor can be approximated as:

$\begin{matrix}{\beta_{w} = {\frac{2\pi}{\lambda_{w}} = {\frac{2\; \pi}{V_{w}\lambda_{o}} = {0.042\mspace{14mu} {m^{- 1}.}}}}} & (71)\end{matrix}$

From Equation (49) the phase delay of the vertical feed line conductoris:

θ_(y)=β_(w) h _(w)≈β_(w) h _(p)=11.640°.  (72)

By adjusting the phase delay of the helical coil so thatθ_(c)=28.974°=40.614°−11.640°, Φ will equal Ψ to match the guidedsurface waveguide mode. To illustrate the relationship between Φ and Ψ,FIG. 11 shows a plot of both over a range of frequencies. As both Φ andΨ are frequency dependent, it can be seen that their respective curvescross over each other at approximately 1.85 MHz.

For a helical coil having a conductor diameter of 0.0881 inches, a coildiameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches,the velocity factor for the coil can be determined using Equation (45)as:

$\begin{matrix}{{V_{f} = {\frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}} = 0.069}},} & (73)\end{matrix}$

and the propagation factor from Equation (35) is:

$\begin{matrix}{\beta_{p} = {\frac{2\; \pi}{V_{f}\lambda_{0}} = {0.564\mspace{14mu} {m^{- 1}.}}}} & (74)\end{matrix}$

With θ_(c)=28.974°, the axial length of the solenoidal helix (H) can bedetermined using Equation (46) such that:

$\begin{matrix}{H = {\frac{\theta_{c}}{\beta_{p}} = {35.2732\mspace{14mu} {{inches}.}}}} & (75)\end{matrix}$

This height determines the location on the helical coil where thevertical feed line conductor is connected, resulting in a coil with8.818 turns (N=H/s).

With the traveling wave phase delay of the coil and vertical feed lineconductor adjusted to match the wave tilt angle (Φ=θ_(c)+θ_(y)=Ψ), theload impedance (Z_(L)) of the charge terminal T₁ can be adjusted forstanding wave resonance of the equivalent image plane model of theguided surface wave probe 300. From the measured permittivity,conductivity and permeability of the earth, the radial propagationconstant can be determined using Equation (57)

γ_(e)=√{square root over (jωu ₁(σ₁ +jωε ₁))}=0.25+j0.292m ⁻¹,  (76)

And the complex depth of the conducting image ground plane can beapproximated from Equation (52) as:

$\begin{matrix}{{{d \approx \frac{2}{\gamma_{e}}} = {3.364 + {j\; 3.963\mspace{14mu} {meters}}}},} & (77)\end{matrix}$

with a corresponding phase shift between the conducting image groundplane and the physical boundary of the earth given by:

θ_(d)=β_(o)(d/2)=4.015−j4.73°.  (78)

Using Equation (65), the impedance seen “looking down” into the lossyconducting medium 303 (i.e., earth) can be determined as:

Z _(in) =Z _(o) tan h(Jθ _(d))=R _(in) +jX _(in)=31.191+j26.27ohms.  (79)

By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 303 with the reactive component (X_(base)) seen“looking up” into the guided surface wave probe 300, the coupling intothe guided surface waveguide mode may be maximized. This can beaccomplished by adjusting the capacitance of the charge terminal T₁without changing the traveling wave phase delays of the coil andvertical feed line conductor. For example, by adjusting the chargeterminal capacitance (C_(T)) to 61.8126 pF, the load impedance fromEquation (62) is:

$\begin{matrix}{{Z_{L} = {\frac{1}{j\; \omega \; C_{T}} = {{- j}\; 1392\mspace{14mu} {ohms}}}},} & (80)\end{matrix}$

and the reactive components at the boundary are matched.

Using Equation (51), the impedance of the vertical feed line conductor(having a diameter (2a) of 0.27 inches) is given as

$\begin{matrix}{{Z_{w} = {{138\; {\log \left( \frac{1.123\; V_{w}\lambda_{0}}{2\; \pi \; a} \right)}} = {537.534\mspace{14mu} {ohms}}}},} & (81)\end{matrix}$

and the impedance seen “looking up” into the vertical feed lineconductor is given by Equation (63) as:

$\begin{matrix}{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{W}\tan \; {h\left( {j\; \theta_{y}} \right)}}}{Z_{W} + {Z_{L}\tan \; {h\left( {j\; \theta_{y}} \right)}}}} = {{- j}\; 835.438\mspace{14mu} {{ohms}.}}}} & (82)\end{matrix}$

Using Equation (47), the characteristic impedance of the helical coil isgiven as

$\begin{matrix}{{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\; {n\left( \frac{V_{f}\lambda_{0}}{D} \right)}} - 1.027} \right\rbrack} = {1446\mspace{14mu} {ohms}}}},} & (83)\end{matrix}$

and the impedance seen “looking up” into the coil at the base is givenby Equation (64) as:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}\tan \; {h\left( {j\; \theta_{c}} \right)}}}{Z_{c} + {Z_{2}\tan \; {h\left( {j\; \theta_{c}} \right)}}}} = {{- j}\; 26.271\mspace{14mu} {{ohms}.}}}} & (84)\end{matrix}$

When compared to the solution of Equation (79), it can be seen that thereactive components are opposite and approximately equal, and thus areconjugates of each other. Thus, the impedance (Z_(ip)) seen “looking up”into the equivalent image plane model of FIGS. 9A and 9B from theperfectly conducting image ground plane is only resistive orZ_(ip)=R+j0.

Referring to FIG. 12, shown is a Smith chart 1200 that graphicallyillustrates an example of the effect of the discontinuous phase jumps onthe impedance (Z_(ip)) seen “looking up” into the equivalent image planemodel of FIG. 9B. First, because of the transition between the chargeterminal and the vertical feed line conductor, the actual load impedanceZ_(L) is normalized with respect to the characteristic impedance (Z_(w))of the vertical feed line conductor and is entered on Smith chart 1200at point 1203 (Z_(L)/Z_(w)). The normalized impedance is thentransferred along the vertical feed line section by an electricaldistance of θ_(y)=β_(w)h_(w)≈β_(w)h_(p) (which is clockwise through anangle of 2θ_(y) on the Smith chart 1200) to point 1206 (Z₂/Z_(w)). Theimpedance at point 1206 is now converted to the actual impedance (Z₂)seen “looking up” into the vertical feed line conductor using Z_(w).

Second, because of the transition between the vertical feed lineconductor and the helical coil, the impedance Z₂ is then normalized withrespect to the characteristic impedance (Z_(c)) of the helical coil.This normalized impedance can now be entered on the Smith chart 1200 atpoint 1209 (Z₂/Z_(c)) and transferred along the helical coiltransmission line section by an electrical distance θ_(c)=β_(p)H, (whichis clockwise through an angle equal to 2θ_(c) on the Smith chart 1200)to point 1212 (Z_(base)/Z_(c)). The jump between point 1206 and point1209 is a result of the discontinuity in the impedance ratios. Theimpedance looking into the base of the coil at point 1212 is thenconverted to the actual impedance (Z_(base)) seen “looking up” into thebase of the coil (or the guided surface wave probe 300) using Z_(c).

Third, because of the transition between the helical coil and the lossyconducting medium, the impedance at Z_(base) is then normalized withrespect to the characteristic impedance (Z_(o)) of the modeled imagespace below the physical boundary of the lossy conducting medium (e.g.,the ground surface). This normalized impedance can now be entered on theSmith chart 1200 at point 1215 (Z_(base)/Z_(o)), and transferred alongthe subsurface image transmission line section by an electrical distanceθ_(d)=β_(o) d/2 (which is clockwise through an angle equal to 2θ_(d) onthe Smith chart 1200) to point 1218 (Z_(ip)/Z_(o)). The jump betweenpoint 1212 and point 1215 is a result of the discontinuity in theimpedance ratios. The impedance looking into the subsurface imagetransmission line at point 1218 is now converted to an actual impedance(Z_(ip)) using Z_(o). When this system is resonated, the impedance atpoint 1218 is Z_(ip)=R_(ip)+j 0. On the Smith chart 1200, Z_(base)/Z_(o)is a larger reactance than Z_(base)/Z_(c). This is because thecharacteristic impedance (Z_(c)) of a helical coil is considerablylarger than the characteristic impedance Z_(o) of free space.

When properly adjusted and tuned, the oscillations on a structure ofsufficient physical height are actually composed of a traveling wave,which is phase delayed to match the angle of the wave tilt associatedwith the lossy conducting medium (Φ=Ψ), plus a standing wave which iselectrically brought into resonance (Z_(ip)=R+j0) by a combination ofthe phase delays of the transmission line sections of the guided surfacewaveguide probe 300 plus the phase discontinuities due to jumps in theratios of the characteristic impedances, as illustrated on the Smithchart 1200 of FIG. 12. The above example illustrates how the threeconsiderations discussed above can be satisfied for launching guidedsurface traveling waves on the lossy conducting medium.

Field strength measurements were carried out to verify the ability ofthe guided surface waveguide probe 300 b (FIG. 7) to couple into aguided surface wave or a transmission line mode. A 70 pF circular platecapacitor was elevated to a height of 16 feet (4.88 meters) and chargedto 30 volts (peak-to-peak) at a frequency of f_(o)=1.85 MHz(λ_(o)=162.162 meters) over soil with constitutive parameters, measuredat f_(o), with a relative permittivity of ε_(r)=15 and a conductivity ofσ₁=0.010 mhos/m. The measured data (documented with a NIST-traceablefield strength meter) is tabulated below in TABLE 1.

TABLE 1 Range Measured FS w/FIM- Predicted FS Percent (miles) 41 (μV/m)(μV/m) Difference 0.6 3400 3415 −0.44% 2 1300 1296 +0.31% 3 840 814+3.19% 4 560 542 +3.32% 5 380 373 +1.88% 6 270 262 +3.05% 7 190 187+1.60% 8 140 134 +4.48% 9 100 97 +3.09% 10 70 71 −1.41%

Referring to FIG. 13, shown is the measured field strength in mV/m(circles) vs. range (in miles) with respect to the theoretical Zennecksurface wave field strengths for 100% and 85% electric charge, as wellas the conventional Norton radiated groundwave predicted for a 16 foottop-loaded vertical mast (monopole at 2.5% radiation efficiency). Thequantity h corresponds to the height of the vertical conducting mast forNorton ground wave radiation with a 55 ohm ground stake. The predictedZenneck fields were calculated from Equation (3), and the standardNorton ground wave was calculated by conventional means. A statisticalanalysis gave a minimized RMS deviation between the measured andtheoretical fields for an electrical efficiency of 97.4%.

When the electric fields produced by a guided surface waveguide probe300 (FIG. 3) are established by matching the traveling wave phase delayof the feed network to the wave tilt angle and the probe structure isresonated with respect to the perfectly conducting image ground plane atcomplex depth z=−d/2, the fields are substantially mode-matched to aguided surface waveguide mode on the surface of the lossy conductingmedium, a guided surface traveling wave is launched along the surface ofthe lossy conducting medium. As illustrated in FIG. 1, the guided fieldstrength curve 103 of the guided electromagnetic field has acharacteristic exponential decay of e^(−αd)/√{square root over (d)} andexhibits a distinctive knee 109 on the log-log scale.

In summary, both analytically and experimentally, the traveling wavecomponent on the structure of the guided surface waveguide probe 300 hasa phase delay (Φ) at its upper terminal that matches the angle (Ψ) ofthe wave tilt of the surface traveling wave (Φ=Ψ). Under this condition,the surface waveguide may be considered to be “mode-matched”.Furthermore, the resonant standing wave component on the structure ofthe guided surface waveguide probe 300 has a V_(MAX) at the chargeterminal T₁ and a V_(MIN) down at the image plane 809 (FIG. 8) whereZ_(ip)=R_(ip)+j 0 at a complex depth of z=−d/2, not at the connection atthe physical boundary 806 of the lossy conducting medium 303 (FIG. 8).Lastly, the charge terminal T₁ is of sufficient height H₁ of FIG. 3(h≥R_(x) tan ψ_(i,B)) so that electromagnetic waves incident onto thelossy conducting medium 303 at the complex Brewster angle do so out at adistance (≥R_(x)) where the 1/√{square root over (r)} term ispredominant. Receive circuits can be utilized with one or more guidedsurface waveguide probes to facilitate wireless transmission and/orpower delivery systems.

Referring next to FIGS. 14A, 14B, 14C and 15, shown are examples ofgeneralized receive circuits for using the surface-guided waves inwireless power delivery systems. FIGS. 14A and 14B-14C include a linearprobe 1403 and a tuned resonator 1406, respectively. FIG. 15 is amagnetic coil 1409 according to various embodiments of the presentdisclosure. According to various embodiments, each one of the linearprobe 1403, the tuned resonator 1406, and the magnetic coil 1409 may beemployed to receive power transmitted in the form of a guided surfacewave on the surface of a lossy conducting medium 303 (FIG. 3) accordingto various embodiments. As mentioned above, in one embodiment the lossyconducting medium 303 comprises a terrestrial medium (or earth).

With specific reference to FIG. 14A, the open-circuit terminal voltageat the output terminals 1413 of the linear probe 1403 depends upon theeffective height of the linear probe 1403. To this end, the terminalpoint voltage may be calculated as

V _(T)=∫₀ ^(h) ^(e) E _(inc) ·dl,  (85)

where E_(inc) is the strength of the incident electric field induced onthe linear probe 1403 in Volts per meter, dl is an element ofintegration along the direction of the linear probe 1403, and h_(e) isthe effective height of the linear probe 1403. An electrical load 1416is coupled to the output terminals 1413 through an impedance matchingnetwork 1419.

When the linear probe 1403 is subjected to a guided surface wave asdescribed above, a voltage is developed across the output terminals 1413that may be applied to the electrical load 1416 through a conjugateimpedance matching network 1419 as the case may be. In order tofacilitate the flow of power to the electrical load 1416, the electricalload 1416 should be substantially impedance matched to the linear probe1403 as will be described below.

Referring to FIG. 14B, a ground current excited coil 1406 a possessing aphase shift equal to the wave tilt of the guided surface wave includes acharge terminal T_(R) that is elevated (or suspended) above the lossyconducting medium 303. The charge terminal T_(R) has a self-capacitanceC_(R). In addition, there may also be a bound capacitance (not shown)between the charge terminal T_(R) and the lossy conducting medium 303depending on the height of the charge terminal T_(R) above the lossyconducting medium 303. The bound capacitance should preferably beminimized as much as is practicable, although this may not be entirelynecessary in every instance of a guided surface waveguide probe 300.

The tuned resonator 1406 a also includes a receiver network comprising acoil L_(R) having a phase shift Φ. One end of the coil L_(R) is coupledto the charge terminal T_(R), and the other end of the coil L_(R) iscoupled to the lossy conducting medium 303. The receiver network caninclude a vertical supply line conductor that couples the coil L_(R) tothe charge terminal T_(R). To this end, the coil 1406 a (which may alsobe referred to as tuned resonator L_(R)-C_(R)) comprises aseries-adjusted resonator as the charge terminal C_(R) and the coilL_(R) are situated in series. The phase delay of the coil 1406 a can beadjusted by changing the size and/or height of the charge terminalT_(R), and/or adjusting the size of the coil L_(R) so that the phase Φof the structure is made substantially equal to the angle of the wavetilt Ψ. The phase delay of the vertical supply line can also be adjustedby, e.g., changing length of the conductor.

For example, the reactance presented by the self-capacitance C_(R) iscalculated as 1/jωC_(R). Note that the total capacitance of thestructure 1406 a may also include capacitance between the chargeterminal T_(R) and the lossy conducting medium 303, where the totalcapacitance of the structure 1406 a may be calculated from both theself-capacitance C_(R) and any bound capacitance as can be appreciated.According to one embodiment, the charge terminal T_(R) may be raised toa height so as to substantially reduce or eliminate any boundcapacitance. The existence of a bound capacitance may be determined fromcapacitance measurements between the charge terminal T_(R) and the lossyconducting medium 303 as previously discussed.

The inductive reactance presented by a discrete-element coil L_(R) maybe calculated as jωL, where L is the lumped-element inductance of thecoil L_(R). If the coil L_(R) is a distributed element, its equivalentterminal-point inductive reactance may be determined by conventionalapproaches. To tune the structure 1406 a, one would make adjustments sothat the phase delay is equal to the wave tilt for the purpose ofmode-matching to the surface waveguide at the frequency of operation.Under this condition, the receiving structure may be considered to be“mode-matched” with the surface waveguide. A transformer link around thestructure and/or an impedance matching network 1423 may be insertedbetween the probe and the electrical load 1426 in order to couple powerto the load. Inserting the impedance matching network 1423 between theprobe terminals 1421 and the electrical load 1426 can effect aconjugate-match condition for maximum power transfer to the electricalload 1426.

When placed in the presence of surface currents at the operatingfrequencies power will be delivered from the surface guided wave to theelectrical load 1426. To this end, an electrical load 1426 may becoupled to the structure 1406 a by way of magnetic coupling, capacitivecoupling, or conductive (direct tap) coupling. The elements of thecoupling network may be lumped components or distributed elements as canbe appreciated.

In the embodiment shown in FIG. 14B, magnetic coupling is employed wherea coil L_(S) is positioned as a secondary relative to the coil L_(R)that acts as a transformer primary. The coil L_(S) may be link-coupledto the coil L_(R) by geometrically winding it around the same corestructure and adjusting the coupled magnetic flux as can be appreciated.In addition, while the receiving structure 1406 a comprises aseries-tuned resonator, a parallel-tuned resonator or even adistributed-element resonator of the appropriate phase delay may also beused.

While a receiving structure immersed in an electromagnetic field maycouple energy from the field, it can be appreciated thatpolarization-matched structures work best by maximizing the coupling,and conventional rules for probe-coupling to waveguide modes should beobserved. For example, a TE₂₀ (transverse electric mode) waveguide probemay be optimal for extracting energy from a conventional waveguideexcited in the TE₂₀ mode. Similarly, in these cases, a mode-matched andphase-matched receiving structure can be optimized for coupling powerfrom a surface-guided wave. The guided surface wave excited by a guidedsurface waveguide probe 300 on the surface of the lossy conductingmedium 303 can be considered a waveguide mode of an open waveguide.Excluding waveguide losses, the source energy can be completelyrecovered. Useful receiving structures may be E-field coupled, H-fieldcoupled, or surface-current excited.

The receiving structure can be adjusted to increase or maximize couplingwith the guided surface wave based upon the local characteristics of thelossy conducting medium 303 in the vicinity of the receiving structure.To accomplish this, the phase delay (Φ) of the receiving structure canbe adjusted to match the angle (Ψ) of the wave tilt of the surfacetraveling wave at the receiving structure. If configured appropriately,the receiving structure may then be tuned for resonance with respect tothe perfectly conducting image ground plane at complex depth z=−d/2.

For example, consider a receiving structure comprising the tunedresonator 1406 a of FIG. 14B, including a coil L_(R) and a verticalsupply line connected between the coil L_(R) and a charge terminalT_(R). With the charge terminal T_(R) positioned at a defined heightabove the lossy conducting medium 303, the total phase shift Φ of thecoil L_(R) and vertical supply line can be matched with the angle (Ψ) ofthe wave tilt at the location of the tuned resonator 1406 a. FromEquation (22), it can be seen that the wave tilt asymptotically passesto

$\begin{matrix}{{W = {{{W}e^{j\; \Psi}} = {\frac{E_{\rho}}{E_{z}}\underset{\rho\rightarrow\infty}{\rightarrow}\frac{1}{\sqrt{ɛ_{r} - {j\frac{\sigma_{1}}{\omega \; ɛ_{0}}}}}}}},} & (86)\end{matrix}$

where ε_(r) comprises the relative permittivity and σ₁ is theconductivity of the lossy conducting medium 303 at the location of thereceiving structure, ε_(o) is the permittivity of free space, and ω=2πf,where f is the frequency of excitation. Thus, the wave tilt angle (Ψ)can be determined from Equation (86).

The total phase shift (Φ=θ_(c)+θ_(y)) of the tuned resonator 1406 aincludes both the phase delay (θ_(c)) through the coil L_(R) and thephase delay of the vertical supply line (θ_(y)). The spatial phase delayalong the conductor length l_(w) of the vertical supply line can begiven by θ_(y)=β_(w)l_(w), where β_(w) is the propagation phase constantfor the vertical supply line conductor. The phase delay due to the coil(or helical delay line) is θ_(c)=β_(p)l_(C), with a physical length ofl_(C) and a propagation factor of

$\begin{matrix}{{\beta_{p} = {\frac{2\; \pi}{\lambda_{p}} = \frac{2\; \pi}{V_{f}\lambda_{0}}}},} & (87)\end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). One or both of thephase delays (θ_(c)+θ_(y)) can be adjusted to match the phase shift Φ tothe angle (Ψ) of the wave tilt. For example, a tap position may beadjusted on the coil L_(R) of FIG. 14B to adjust the coil phase delay(θ_(c)) to match the total phase shift to the wave tilt angle (Φ=Ψ). Forexample, a portion of the coil can be bypassed by the tap connection asillustrated in FIG. 14B. The vertical supply line conductor can also beconnected to the coil L_(R) via a tap, whose position on the coil may beadjusted to match the total phase shift to the angle of the wave tilt.

Once the phase delay (Φ) of the tuned resonator 1406 a has beenadjusted, the impedance of the charge terminal T_(R) can then beadjusted to tune to resonance with respect to the perfectly conductingimage ground plane at complex depth z=−d/2. This can be accomplished byadjusting the capacitance of the charge terminal T₁ without changing thetraveling wave phase delays of the coil L_(R) and vertical supply line.The adjustments are similar to those described with respect to FIGS. 9Aand 9B and the Smith chart of FIG. 12.

The impedance seen “looking down” into the lossy conducting medium 303to the complex image plane is given by:

Z _(in) =R _(in) +jX _(in) =Z _(o) tan h(jβ _(o)(d/2)),  (88)

where β_(o)=ω√{square root over (μ_(o)ε_(o))}. For vertically polarizedsources over the earth, the depth of the complex image plane can begiven by:

d/2≈1/√{square root over (jωμ ₁σ₁−ω²μ₁ε₁)},  (89)

where μ₁ is the permeability of the lossy conducting medium 303 andε₁=ε_(r)ε_(o).

At the base of the tuned resonator 1406 a, the impedance seen “lookingup” into the receiving structure is Z_(↑)=Z_(base) as illustrated inFIG. 9A. With a terminal impedance of:

$\begin{matrix}{{Z_{R} = \frac{1}{j\; \omega \; C_{R}}},} & (90)\end{matrix}$

where C_(R) is the self-capacitance of the charge terminal T_(R), theimpedance seen “looking up” into the vertical supply line conductor ofthe tuned resonator 1406 a is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{R} + {Z_{W}\tan \; {h\left( {j\; \beta_{w}h_{w}} \right)}}}{Z_{W} + {Z_{R}\tan \; {h\left( {j\; \beta_{w}h_{w}} \right)}}}} = {Z_{W}\frac{Z_{2} + {Z_{c}\tan \; {h\left( {j\; \theta_{y}} \right)}}}{Z_{c} + {Z_{2}\tan \; {h\left( {j\; \theta_{y}} \right)}}}}}},} & (91)\end{matrix}$

and the impedance seen “looking up” into the coil L_(R) of the tunedresonator 1406 a is given by:

$\begin{matrix}{Z_{base} = {{R_{base} + {jX}_{base}} = {{Z_{R}\frac{Z_{2} + {Z_{R}\tan \; {h\left( {j\; \beta_{p}H} \right)}}}{Z_{R} + {Z_{2}\tan \; {h\left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{R}\tan \; {h\left( {j\; \theta_{c}} \right)}}}{Z_{R} + {Z_{2}\tan \; {h\left( {j\; \theta_{c}} \right)}}}.}}}}} & (92)\end{matrix}$

By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 303 with the reactive component (X_(base)) seen“looking up” into the tuned resonator 1406 a, the coupling into theguided surface waveguide mode may be maximized.

Referring next to FIG. 14C, shown is an example of a tuned resonator1406 b that does not include a charge terminal T_(R) at the top of thereceiving structure. In this embodiment, the tuned resonator 1406 b doesnot include a vertical supply line coupled between the coil L_(R) andthe charge terminal T_(R). Thus, the total phase shift (Φ) of the tunedresonator 1406 b includes only the phase delay (θ_(c)) through the coilL_(R). As with the tuned resonator 1406 a of FIG. 14B, the coil phasedelay θ_(c) can be adjusted to match the angle (Ψ) of the wave tiltdetermined from Equation (86), which results in Φ=Ψ. While powerextraction is possible with the receiving structure coupled into thesurface waveguide mode, it is difficult to adjust the receivingstructure to maximize coupling with the guided surface wave without thevariable reactive load provided by the charge terminal T_(R).

Referring to FIG. 14D, shown is a flow chart illustrating an example ofadjusting a receiving structure to substantially mode-match to a guidedsurface waveguide mode on the surface of the lossy conducting medium303. Beginning with 1453, if the receiving structure includes a chargeterminal T_(R) (e.g., of the tuned resonator 1406 a of FIG. 14B), thenthe charge terminal T_(R) is positioned at a defined height above alossy conducting medium 303 at 1456. As the surface guided wave has beenestablished by a guided surface waveguide probe 300, the physical height(h_(p)) of the charge terminal T_(R) height may be below that of theeffective height. The physical height may be selected to reduce orminimize the bound charge on the charge terminal T_(R) (e.g., four timesthe spherical diameter of the charge terminal). If the receivingstructure does not include a charge terminal T_(R) (e.g., of the tunedresonator 1406 b of FIG. 14C), then the flow proceeds to 1459.

At 1459, the electrical phase delay Φ of the receiving structure ismatched to the complex wave tilt angle Ψ defined by the localcharacteristics of the lossy conducting medium 303. The phase delay(θ_(c)) of the helical coil and/or the phase delay (θ_(y)) of thevertical supply line can be adjusted to make Φ equal to the angle (Ψ) ofthe wave tilt (W). The angle (Ψ) of the wave tilt can be determined fromEquation (86). The electrical phase Φ can then be matched to the angleof the wave tilt. For example, the electrical phase delay Φ=θ_(c)+θ_(y)can be adjusted by varying the geometrical parameters of the coil L_(R)and/or the length (or height) of the vertical supply line conductor.

Next at 1462, the load impedance of the charge terminal T_(R) can betuned to resonate the equivalent image plane model of the tunedresonator 1406 a. The depth (d/2) of the conducting image ground plane809 (FIG. 9A) below the receiving structure can be determined usingEquation (89) and the values of the lossy conducting medium 303 (e.g.,the earth) at the receiving structure, which can be locally measured.Using that complex depth, the phase shift (θ_(d)) between the imageground plane 809 and the physical boundary 806 (FIG. 9A) of the lossyconducting medium 303 can be determined using θ_(d)=β_(o) d/2. Theimpedance (Z_(in)) as seen “looking down” into the lossy conductingmedium 303 can then be determined using Equation (88). This resonancerelationship can be considered to maximize coupling with the guidedsurface waves.

Based upon the adjusted parameters of the coil L_(R) and the length ofthe vertical supply line conductor, the velocity factor, phase delay,and impedance of the coil L_(R) and vertical supply line can bedetermined. In addition, the self-capacitance (C_(R)) of the chargeterminal T_(R) can be determined using, e.g., Equation (24). Thepropagation factor (β_(p)) of the coil L_(R) can be determined usingEquation (87) and the propagation phase constant (β_(w)) for thevertical supply line can be determined using Equation (49). Using theself-capacitance and the determined values of the coil L_(R) andvertical supply line, the impedance (Z_(base)) of the tuned resonator1406 a as seen “looking up” into the coil L_(R) can be determined usingEquations (90), (91), and (92).

The equivalent image plane model of FIG. 9A also applies to the tunedresonator 1406 a of FIG. 14B. The tuned resonator 1406 a can be tuned toresonance with respect to the complex image plane by adjusting the loadimpedance Z_(R) of the charge terminal T_(R) such that the reactancecomponent X_(base) of Z_(base) cancels out the reactance component ofX_(in) of Z_(in), or X_(base)+X_(in)=0. Thus, the impedance at thephysical boundary 806 (FIG. 9A) “looking up” into the coil of the tunedresonator 1406 a is the conjugate of the impedance at the physicalboundary 806 “looking down” into the lossy conducting medium 303. Theload impedance Z_(R) can be adjusted by varying the capacitance (C_(R))of the charge terminal T_(R) without changing the electrical phase delayΦ=θ_(c)+θ_(y) seen by the charge terminal T_(R). An iterative approachmay be taken to tune the load impedance Z_(R) for resonance of theequivalent image plane model with respect to the conducting image groundplane 809. In this way, the coupling of the electric field to a guidedsurface waveguide mode along the surface of the lossy conducting medium303 (e.g., earth) can be improved and/or maximized.

Referring to FIG. 15, the magnetic coil 1409 comprises a receive circuitthat is coupled through an impedance matching network 1433 to anelectrical load 1436. In order to facilitate reception and/or extractionof electrical power from a guided surface wave, the magnetic coil 1409may be positioned so that the magnetic flux of the guided surface wave,H_(φ), passes through the magnetic coil 1409, thereby inducing a currentin the magnetic coil 1409 and producing a terminal point voltage at itsoutput terminals 1429. The magnetic flux of the guided surface wavecoupled to a single turn coil is expressed by

=∫∫_(A) _(CS) μ_(r)μ_(o) {right arrow over (H)}·{circumflex over(n)}dA  (93)

where

is the coupled magnetic flux, μ_(r) is the effective relativepermeability of the core of the magnetic coil 1409, μ_(o) is thepermeability of free space, H is the incident magnetic field strengthvector, {circumflex over (n)} is a unit vector normal to thecross-sectional area of the turns, and A_(CS) is the area enclosed byeach loop. For an N-turn magnetic coil 1409 oriented for maximumcoupling to an incident magnetic field that is uniform over thecross-sectional area of the magnetic coil 1409, the open-circuit inducedvoltage appearing at the output terminals 1429 of the magnetic coil 1409is

$\begin{matrix}{{V = {{{- N}\frac{d\; }{dt}} \approx {{- j}\; \omega \; \mu_{r}\mu_{0}{NHA}_{CS}}}},} & (94)\end{matrix}$

where the variables are defined above. The magnetic coil 1409 may betuned to the guided surface wave frequency either as a distributedresonator or with an external capacitor across its output terminals1429, as the case may be, and then impedance-matched to an externalelectrical load 1436 through a conjugate impedance matching network1433.

Assuming that the resulting circuit presented by the magnetic coil 1409and the electrical load 1436 are properly adjusted and conjugateimpedance matched, via impedance matching network 1433, then the currentinduced in the magnetic coil 1409 may be employed to optimally power theelectrical load 1436. The receive circuit presented by the magnetic coil1409 provides an advantage in that it does not have to be physicallyconnected to the ground.

With reference to FIGS. 14A, 14B, 14C and 15, the receive circuitspresented by the linear probe 1403, the mode-matched structure 1406, andthe magnetic coil 1409 each facilitate receiving electrical powertransmitted from any one of the embodiments of guided surface waveguideprobes 300 described above. To this end, the energy received may be usedto supply power to an electrical load 1416/1426/1436 via a conjugatematching network as can be appreciated. This contrasts with the signalsthat may be received in a receiver that were transmitted in the form ofa radiated electromagnetic field. Such signals have very low availablepower and receivers of such signals do not load the transmitters.

It is also characteristic of the present guided surface waves generatedusing the guided surface waveguide probes 300 described above that thereceive circuits presented by the linear probe 1403, the mode-matchedstructure 1406, and the magnetic coil 1409 will load the excitationsource 312 (FIG. 3) that is applied to the guided surface waveguideprobe 300, thereby generating the guided surface wave to which suchreceive circuits are subjected. This reflects the fact that the guidedsurface wave generated by a given guided surface waveguide probe 300described above comprises a transmission line mode. By way of contrast,a power source that drives a radiating antenna that generates a radiatedelectromagnetic wave is not loaded by the receivers, regardless of thenumber of receivers employed.

Thus, together one or more guided surface waveguide probes 300 and oneor more receive circuits in the form of the linear probe 1403, the tunedmode-matched structure 1406, and/or the magnetic coil 1409 can togethermake up a wireless distribution system. Given that the distance oftransmission of a guided surface wave using a guided surface waveguideprobe 300 as set forth above depends upon the frequency, it is possiblethat wireless power distribution can be achieved across wide areas andeven globally.

The conventional wireless-power transmission/distribution systemsextensively investigated today include “energy harvesting” fromradiation fields and also sensor coupling to inductive or reactivenear-fields. In contrast, the present wireless-power system does notwaste power in the form of radiation which, if not intercepted, is lostforever. Nor is the presently disclosed wireless-power system limited toextremely short ranges as with conventional mutual-reactance couplednear-field systems. The wireless-power system disclosed hereinprobe-couples to the novel surface-guided transmission line mode, whichis equivalent to delivering power to a load by a waveguide or a loaddirectly wired to the distant power generator. Not counting the powerrequired to maintain transmission field strength plus that dissipated inthe surface waveguide, which at extremely low frequencies isinsignificant relative to the transmission losses in conventionalhigh-tension power lines at 60 Hz, all of the generator power goes onlyto the desired electrical load. When the electrical load demand isterminated, the source power generation is relatively idle.

Referring next to FIG. 16A shown is a schematic that represents thelinear probe 1403 and the mode-matched structure 1406. FIG. 16B shows aschematic that represents the magnetic coil 1409. The linear probe 1403and the mode-matched structure 1406 may each be considered a Theveninequivalent represented by an open-circuit terminal voltage source V_(S)and a dead network terminal point impedance Z_(S). The magnetic coil1409 may be viewed as a Norton equivalent represented by a short-circuitterminal current source I_(S) and a dead network terminal pointimpedance Z_(S). Each electrical load 1416/1426/1436 (FIGS. 14A, 14B and15) may be represented by a load impedance Z_(L). The source impedanceZ_(S) comprises both real and imaginary components and takes the formZ_(S)=R_(S)+jX_(S).

According to one embodiment, the electrical load 1416/1426/1436 isimpedance matched to each receive circuit, respectively. Specifically,each electrical load 1416/1426/1436 presents through a respectiveimpedance matching network 1419/1423/1433 a load on the probe networkspecified as Z_(L)′ expressed as Z_(L)′=R_(L)′+j X_(L)′, which will beequal to Z_(L)′=Z_(s)*=R_(S)−j X_(S), where the presented load impedanceZ_(L)′ is the complex conjugate of the actual source impedance Z_(S).The conjugate match theorem, which states that if, in a cascadednetwork, a conjugate match occurs at any terminal pair then it willoccur at all terminal pairs, then asserts that the actual electricalload 1416/1426/1436 will also see a conjugate match to its impedance,Z_(L)′. See Everitt, W. L. and G. E. Anner, Communication Engineering,McGraw-Hill, 3^(rd) edition, 1956, p. 407. This ensures that therespective electrical load 1416/1426/1436 is impedance matched to therespective receive circuit and that maximum power transfer isestablished to the respective electrical load 1416/1426/1436.

Operation of a guided surface waveguide probe 300 may be controlled toadjust for variations in operational conditions associated with theguided surface waveguide probe 300. For example, an adaptive probecontrol system 321 (FIG. 3) can be used to control the feed network 309and/or the charge terminal T₁ to control the operation of the guidedsurface waveguide probe 300. Operational conditions can include, but arenot limited to, variations in the characteristics of the lossyconducting medium 303 (e.g., conductivity σ and relative permittivityε_(r)), variations in field strength and/or variations in loading of theguided surface waveguide probe 300. As can be seen from Equations (31),(41) and (42), the index of refraction (n), the complex Brewster angle(θ_(i,B)), and the wave tilt (|W|e^(jΨ)) can be affected by changes insoil conductivity and permittivity resulting from, e.g., weatherconditions.

Equipment such as, e.g., conductivity measurement probes, permittivitysensors, ground parameter meters, field meters, current monitors and/orload receivers can be used to monitor for changes in the operationalconditions and provide information about current operational conditionsto the adaptive probe control system 321. The probe control system 321can then make one or more adjustments to the guided surface waveguideprobe 300 to maintain specified operational conditions for the guidedsurface waveguide probe 300. For instance, as the moisture andtemperature vary, the conductivity of the soil will also vary.Conductivity measurement probes and/or permittivity sensors may belocated at multiple locations around the guided surface waveguide probe300. Generally, it would be desirable to monitor the conductivity and/orpermittivity at or about the Hankel crossover distance R_(x) for theoperational frequency. Conductivity measurement probes and/orpermittivity sensors may be located at multiple locations (e.g., in eachquadrant) around the guided surface waveguide probe 300.

FIG. 17A shows an example of a conductivity measurement probe that canbe installed for monitoring changes in soil conductivity. As shown inFIG. 17A, a series of measurement probes are inserted along a straightline in the soil. For example, the probes may be 9/16-inch diameter rodswith a penetration depth of 12 inches or more, and spaced apart by d=18inches. DS1 is a 100 Watt light bulb and R1 is a 5 Watt, 14.6 Ohmresistance. By applying an AC voltage to the circuit and measuring V1across the resistance and V2 across the center probes, the conductivitycan be determined by the weighted ratio of σ=21(V1/V2). The measurementscan be filtered to obtain measurements related only to the AC voltagesupply frequency. Different configurations using other voltages,frequencies, probe sizes, depths and/or spacing may also be utilized.

Open wire line probes can also be used to measure conductivity andpermittivity of the soil. As illustrated in FIG. 17B, impedance ismeasured between the tops of two rods inserted into the soil (lossymedium) using, e.g., an impedance analyzer. If an impedance analyzer isutilized, measurements (R+jX) can be made over a range of frequenciesand the conductivity and permittivity determined from the frequencydependent measurements using

$\begin{matrix}{{\sigma = {{{\frac{8.84}{C_{o}}\left\lbrack \frac{R}{R^{2} + X^{2}} \right\rbrack}\mspace{14mu} {and}\mspace{14mu} ɛ_{r}} = {\frac{10^{6}}{2\; \pi \; {fC}_{0}}\left\lbrack \frac{R}{R^{2} + X^{2}} \right\rbrack}}},} & (95)\end{matrix}$

where C₀ is the capacitance in pF of the probe in air.

The conductivity measurement probes and/or permittivity sensors can beconfigured to evaluate the conductivity and/or permittivity on aperiodic basis and communicate the information to the probe controlsystem 321 (FIG. 3). The information may be communicated to the probecontrol system 321 through a network such as, but not limited to, a LAN,WLAN, cellular network, or other appropriate wired or wirelesscommunication network. Based upon the monitored conductivity and/orpermittivity, the probe control system 321 may evaluate the variation inthe index of refraction (n), the complex Brewster angle (θ_(i,B)),and/or the wave tilt (|W|e^(jΨ)) and adjust the guided surface waveguideprobe 300 to maintain the phase delay (Φ) of the feed network 309 equalto the wave tilt angle (Ψ) and/or maintain resonance of the equivalentimage plane model of the guided surface waveguide probe 300. This can beaccomplished by adjusting, e.g., θ_(y), θ_(c) and/or C_(T). Forinstance, the probe control system 321 can adjust the self-capacitanceof the charge terminal T₁ or the phase delay (θ_(y), θ_(c)) applied tothe charge terminal T₁ to maintain the electrical launching efficiencyof the guided surface wave at or near its maximum. The phase applied tothe charge terminal T₁ can be adjusted by varying the tap position onthe coil 709, and/or by including a plurality of predefined taps alongthe coil 709 and switching between the different predefined taplocations to maximize the launching efficiency.

Field or field strength (FS) meters (e.g., a FIM-41 FS meter, PotomacInstruments, Inc., Silver Spring, Md.) may also be distributed about theguided surface waveguide probe 300 to measure field strength of fieldsassociated with the guided surface wave. The field or FS meters can beconfigured to detect the field strength and/or changes in the fieldstrength (e.g., electric field strength) and communicate thatinformation to the probe control system 321. The information may becommunicated to the probe control system 321 through a network such as,but not limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. As the load and/or environmental conditionschange or vary during operation, the guided surface waveguide probe 300may be adjusted to maintain specified field strength(s) at the FS meterlocations to ensure appropriate power transmission to the receivers andthe loads they supply.

For example, the phase delay (Φ=θ_(y)+θ_(c)) applied to the chargeterminal T₁ can be adjusted to match the wave tilt angle (Ψ). Byadjusting one or both phase delays, the guided surface waveguide probe300 can be adjusted to ensure the wave tilt corresponds to the complexBrewster angle. This can be accomplished by adjusting a tap position onthe coil 709 (FIG. 7) to change the phase delay supplied to the chargeterminal T₁. The voltage level supplied to the charge terminal T₁ canalso be increased or decreased to adjust the electric field strength.This may be accomplished by adjusting the output voltage of theexcitation source 312 (FIG. 3) or by adjusting or reconfiguring the feednetwork 309 (FIG. 3). For instance, the position of the tap 724 (FIG. 7)for the AC source 712 (FIG. 7) can be adjusted to increase the voltageseen by the charge terminal T₁. Maintaining field strength levels withinpredefined ranges can improve coupling by the receivers, reduce groundcurrent losses, and avoid interference with transmissions from otherguided surface waveguide probes 300.

Referring to FIG. 18, shown is an example of an adaptive control system330 including the probe control system 321 of FIG. 3, which isconfigured to adjust the operation of a guided surface waveguide probe300, based upon monitored conditions. As in FIGS. 3 and 7, an AC source712 acts as the excitation source (312 of FIG. 3) for the chargeterminal T₁. The AC source 712 is coupled to the guided surfacewaveguide probe 400 d through a feed network (309 of FIG. 3) comprisinga coil 709. The AC source 712 can be connected across a lower portion ofthe coil 709 through a tap 724, as shown in FIG. 7, or can beinductively coupled to the coil 709 by way of a primary coil. The coil709 can be coupled to a ground stake 715 (FIG. 7) at a first end and thecharge terminal T₁ at a second end. In some implementations, theconnection to the charge terminal T₁ can be adjusted using a tap 721(FIG. 7) at the second end of the coil 709. An ammeter located betweenthe coil 709 and ground stake 715 can be used to provide an indicationof the magnitude of the current flow (I₀) at the base of the guidedsurface waveguide probe 300. Alternatively, a current clamp may be usedaround the conductor coupled to the ground stake 715 to obtain anindication of the magnitude of the current flow (I₀).

The probe control system 321 can be implemented with hardware, firmware,software executed by hardware, or a combination thereof. For example,the probe control system 321 can include processing circuitry includinga processor and a memory, both of which can be coupled to a localinterface such as, for example, a data bus with an accompanyingcontrol/address bus as can be appreciated by those with ordinary skillin the art. A probe control application may be executed by the processorto adjust the operation of the guided surface waveguide probe 400 basedupon monitored conditions. The probe control system 321 can also includeone or more network interfaces for communicating with the variousmonitoring devices. Communications can be through a network such as, butnot limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. The probe control system 321 may comprise, forexample, a computer system such as a server, desktop computer, laptop,or other system with like capability.

The adaptive control system 330 can include one or more ground parametermeter(s) 333 such as, but not limited to, a conductivity measurementprobe of FIG. 17A and/or an open wire probe of FIG. 17B. The groundparameter meter(s) 333 can be distributed about the guided surfacewaveguide probe 300 at, e.g., about the Hankel crossover distance(R_(x)) associated with the probe operating frequency. For example, anopen wire probe of FIG. 17B may be located in each quadrant around theguided surface waveguide probe 300 to monitor the conductivity andpermittivity of the lossy conducting medium as previously described. Theground parameter meter(s) 333 can be configured to determine theconductivity and permittivity of the lossy conducting medium on aperiodic basis and communicate the information to the probe controlsystem 321 for potential adjustment of the guided surface waveguideprobe 300. In some cases, the ground parameter meter(s) 333 maycommunicate the information to the probe control system 321 only when achange in the monitored conditions is detected.

The adaptive control system 330 can also include one or more fieldmeter(s) 336 such as, but not limited to, an electric field strength(FS) meter. The field meter(s) 336 can be distributed about the guidedsurface waveguide probe 300 beyond the Hankel crossover distance (R_(x))where the guided field strength curve 103 (FIG. 1) dominates theradiated field strength curve 106 (FIG. 1). For example, a plurality offiled meters 336 may be located along one or more radials extendingoutward from the guided surface waveguide probe 300 to monitor theelectric field strength as previously described. The field meter(s) 336can be configured to determine the field strength on a periodic basisand communicate the information to the probe control system 321 forpotential adjustment of the guided surface waveguide probe 300. In somecases, the field meter(s) 336 may communicate the information to theprobe control system 321 only when a change in the monitored conditionsis detected.

Other variables can also be monitored and used to adjust the operationof the guided surface waveguide probe 300. For instance, the groundcurrent flowing through the ground stake 715 (FIG. 7) can be used tomonitor the operation of the guided surface waveguide probe 300. Forexample, the ground current can provide an indication of changes in theloading of the guided surface waveguide probe 300 and/or the coupling ofthe electric field into the guided surface wave mode on the surface ofthe lossy conducting medium 303. Real power delivery may be determinedby monitoring the AC source 712 (or excitation source 312 of FIG. 3). Insome implementations, the guided surface waveguide probe 300 may beadjusted to maximize coupling into the guided surface waveguide modebased at least in part upon the current indication. By adjusting thephase delay (Φ=θ_(y)+θ_(c)) supplied to the charge terminal T₁, thematch with the wave tilt angle (Ψ) can be maintained for illumination atthe complex Brewster angle for guided surface wave transmissions in thelossy conducting medium 303 (e.g., the earth). This can be accomplishedby adjusting the tap position on the coil 709. However, the groundcurrent can also be affected by receiver loading. If the ground currentis above the expected current level, then this may indicate thatunaccounted loading of the guided surface waveguide probe 400 is takingplace.

The excitation source 312 (or AC source 712) can also be monitored toensure that overloading does not occur. As real load on the guidedsurface waveguide probe 300 increases, the output voltage of theexcitation source 312, or the voltage supplied to the charge terminal T₁from the coil, can be increased to increase field strength levels,thereby avoiding additional load currents. In some cases, the receiversthemselves can be used as sensors monitoring the condition of the guidedsurface waveguide mode. For example, the receivers can monitor fieldstrength and/or load demand at the receiver. The receivers can beconfigured to communicate information about current operationalconditions to the probe control system 321. The information may becommunicated to the probe control system 321 through a network such as,but not limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. Based upon the information, the probe controlsystem 321 can then adjust the guided surface waveguide probe 300 forcontinued operation. For example, the phase delay (Φ=θ_(y)+θ_(c))applied to the charge terminal T₁ can be adjusted to maintain theelectrical launching efficiency of the guided surface waveguide probe300, to supply the load demands of the receivers. In some cases, theprobe control system 321 may adjust the guided surface waveguide probe300 to reduce loading on the excitation source 312 and/or guided surfacewaveguide probe 300. For example, the voltage supplied to the chargeterminal T₁ may be reduced to lower field strength and prevent couplingto a portion of the most distant load devices.

The guided surface waveguide probe 300 can be adjusted by the probecontrol system 321 using, e.g., one or more tap controllers 339. In FIG.18, the connection from the coil 709 to the upper charge terminal T₁ iscontrolled by a tap controller 339. In response to a change in themonitored conditions (e.g., a change in conductivity, permittivity,and/or electric field strength), the probe control system cancommunicate a control signal to the tap controller 339 to initiate achange in the tap position. The tap controller 339 can be configured tovary the tap position continuously along the coil 709 or incrementallybased upon predefined tap connections. The control signal can include aspecified tap position or indicate a change by a defined number of tapconnections. By adjusting the tap position, the phase delay (Φ) of thecharge terminal T₁ can be adjusted to maintain and/or improve couplingof the guided surface waveguide mode.

The guided surface waveguide probe 300 can also be adjusted by the probecontrol system 321 using, e.g., a charge terminal control system 348. Byadjusting the impedance of the charge terminal T₁, it is possible toadjust the coupling into the guided surface waveguide mode. The chargeterminal control system 348 can be configured to change the capacitanceof the charge terminal T₁. By adjusting the load impedance Z_(L) of thecharge terminal T₁ while maintaining Φ=Ψ, resonance with respect to theconductive image ground plane can be maintained. In this way, couplingof the electric field to a guided surface waveguide mode along thesurface of the lossy conducting medium 303 (e.g., earth) can be improvedand/or maximized.

As has been discussed, the probe control system 321 of the adaptivecontrol system 330 can monitor the operating conditions of the guidedsurface waveguide probe 300 by communicating with one or more remotelylocated monitoring devices such as, but not limited to, a groundparameter meter 333 and/or a field meter 336. The probe control system321 can also monitor other conditions by accessing information from,e.g., the AC source 712 (or excitation source 312). Based upon themonitored information, the probe control system 321 can determine ifadjustment of the guided surface waveguide probe 300 is needed toimprove and/or maximize the launching efficiency. In response to achange in one or more of the monitored conditions, the probe controlsystem 321 can initiate an adjustment of one or more of the phase delay(θ_(p), θ_(c)) applied to the charge terminal T₁ and/or the loadimpedance Z_(L) of the charge terminal T₁. In some implantations, theprobe control system 321 can evaluate the monitored conditions toidentify the source of the change. If the monitored condition(s) wascaused by a change in receiver load, then adjustment of the guidedsurface waveguide probe 300 may be avoided. If the monitoredcondition(s) affect the launching efficiency of the guided surfacewaveguide probe 400, then the probe control system 321 can initiateadjustments of the guided surface waveguide probe 300 to improve and/ormaximize the launching efficiency.

In some embodiments, the size of the charge terminal T₁ can be adjustedto control the load impedance Z_(L) of the guided surface waveguideprobe 300. For example, the self-capacitance of the charge terminal T₁can be varied by changing the size of the terminal. The chargedistribution can also be improved by increasing the size of the chargeterminal T₁, which can reduce the chance of an electrical discharge fromthe charge terminal T₁. In other embodiments, the charge terminal T₁ caninclude a variable inductance that can be adjusted to change the loadimpedance Z_(L). Control of the charge terminal T₁ size can be providedby the probe control system 321 through the charge terminal controlsystem 348 or through a separate control system.

FIGS. 19A and 19B illustrate an example of a variable terminal 203 thatcan be used as a charge terminal T₁ of the guided surface waveguideprobe 300 or a charge terminal T_(R) of the tuned resonator 1406 (FIGS.14B and 14C). For example, the variable terminal 203 can include aninner cylindrical section 206 nested inside of an outer cylindricalsection 209. The inner and outer cylindrical sections 206 and 209 caninclude plates across the bottom and top, respectively. In FIG. 19A, thecylindrically shaped variable terminal 203 is shown in a contractedcondition having a first size, which can be associated with a firsteffective spherical diameter. To change the size of the terminal, andthus the effective spherical diameter, one or both sections of thevariable terminal 203 can be extended to increase the surface area asshown in FIG. 19B. This may be accomplished by using a driving mechanismsuch as an electric motor or hydraulic cylinder that is electricallyisolated to prevent discharge of the charge on the terminal. In thisway, the capacitance (C₁ or C_(R)) of the charge terminal T₁ or T_(R),and thus the load impedance (Z_(L) or Z_(R)) of the charge terminal T₁or T_(R), can be adjusted.

Referring next to FIG. 20, shown is a schematic representationillustrating a variable terminal 212 including a variable inductance 215within the outer surface 218 of the terminal 212. By placing thevariable inductor within the terminal 212, the load impedance Z_(L) ofthe guided surface waveguide probe 300 of FIG. 3 (or the load impedanceZ_(R) of the tuned resonator 1406 of FIGS. 14B and 14C) can be adjustedby adjusting the inductance 215, without affecting the charge surface ofthe charge terminal T₁. In some embodiments, the variable terminal 203of FIGS. 19A and 19B can include a variable inductance 215 within thecylindrical sections 206 and 209. Such a combination can provide a widerrange of control over the load impedance Z_(L) of the guided surfacewaveguide probe 300.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations setforth for a clear understanding of the principles of the disclosure.Many variations and modifications may be made to the above-describedembodiment(s) without departing substantially from the spirit andprinciples of the disclosure. All such modifications and variations areintended to be included herein within the scope of this disclosure andprotected by the following claims. In addition, all optional andpreferred features and modifications of the described embodiments anddependent claims are usable in all aspects of the disclosure taughtherein. Furthermore, the individual features of the dependent claims, aswell as all optional and preferred features and modifications of thedescribed embodiments are combinable and interchangeable with oneanother.

Therefore, the following is claimed:
 1. A method, comprising: coupling areceiving structure to a lossy conducting medium; and mode-matching thereceiving structure with a guided surface wave established on the lossyconducting medium, where a traveling wave phase delay (Φ) of thereceiving structure is matched to a wave tilt angle (Ψ) associated withthe guided surface wave, the wave tilt angle (Ψ) based at least in partupon characteristics of the lossy conducting medium in a vicinity of thereceiving structure.
 2. The method of claim 1, comprising suspending acharge terminal of the receiving structure at a defined height over asurface of the lossy conducting medium.
 3. The method of claim 2,wherein the receiving structure comprises a receiver network coupledbetween the charge terminal and the lossy conducting medium.
 4. Themethod of claim 3, wherein the receiver network comprises a coil coupledto the lossy conducting medium and a supply line conductor coupledbetween the coil and the charge terminal, where the traveling wave phasedelay (Φ) is based upon a phase delay (θ_(c)) of the coil and a phasedelay (θ_(y)) of the supply line conductor.
 5. The method of claim 4,wherein adjusting the traveling wave phase delay (Φ) comprises adjustinga position of a tap on the coil to vary the phase delay (θ_(c)) of thecoil.
 6. The method of claim 5, wherein the supply line conductor iscoupled to the coil via the tap.
 7. The method of claim 2, wherein thecharge terminal has an effective spherical diameter, and the definedheight of the charge terminal is at least four times the effectivespherical diameter to reduce bound capacitance.
 8. The method of claim2, comprising resonating the receiving structure relative to an imageplane at a complex depth below the surface of the lossy conductingmedium.
 9. The method of claim 8, wherein resonating the receivingstructure comprises adjusting a load impedance (Z_(L)) of the chargeterminal based upon an image ground plane impedance (Z_(in)) associatedwith the lossy conducting medium.
 10. The method of claim 8, whereinresonating the receiving structure establishes a standing wave on thereceiving structure by exploiting phase delays from transmission linesections of the receiving structure plus phase jumps arising fromdiscontinuities in characteristic impedances of the transmission linesections, the standing wave superposed with a traveling wave on thereceiving structure.
 11. The method of claim 1, comprising extractingelectrical power from the receiving structure via a coil.
 12. Areceiving structure for mode-matching with a guided surface waveestablished on a lossy conducting medium, the receiving structurecomprising: a charge terminal elevated over the lossy conducting medium;and a receiver network coupled between the charge terminal and the lossyconducting medium, the receiver network having a phase delay (Φ) thatmatches a wave tilt angle (Ψ) associated with the guided surface wave,the wave tilt angle (Ψ) based at least in part upon characteristics ofthe lossy conducting medium in a vicinity of the receiving structure.13. The receiving structure of claim 12, wherein the charge terminal hasa variable load impedance (Z_(L)).
 14. The receiving structure of claim13, wherein the variable load impedance (Z_(L)) is determined based uponan image ground plane impedance (Z_(in)) associated with the lossyconducting medium in the vicinity of the receiving structure.
 15. Thereceiving structure of claim 14, wherein the load impedance (Z_(L)) isadjusted to resonate the receiving structure relative to an image planeat a complex depth below a surface of the lossy conducting medium. 16.The receiving structure of claim 15, wherein resonating the receivingstructure establishes a standing wave on the receiving structure byexploiting phase delays from transmission line sections of the receivernetwork plus phase jumps arising from discontinuities in characteristicimpedances of the transmission line sections.
 17. The receivingstructure of claim 12, wherein the receiver network comprises a coilcoupled to the lossy conducting medium and a supply line conductorcoupled between the coil and the charge terminal, where the phase delay(Φ) of the receiver network is based upon a phase delay (θ_(c)) of thecoil and a phase delay (θ_(y)) of the supply line conductor.
 18. Thereceiving structure of claim 17, further comprising a variable tapconfigured to adjust the phase delay (θ_(c)) of the coil.
 19. Thereceiving structure of claim 12, comprising an impedance matchingnetwork coupled to a coil.
 20. The receiving structure of claim 19,wherein the impedance matching network is inductively coupled to thecoil.
 21. A method, comprising: positioning a receive structure relativeto a terrestrial medium; and receiving, via the receive structure,energy conveyed in a form of a guided surface wave on a surface of theterrestrial medium.
 22. The method of claim 21, wherein the receivestructure loads an excitation source coupled to a guided surfacewaveguide probe that generates the guided surface wave.
 23. The methodof claim 21, wherein the energy comprises electrical power, and themethod further comprises applying the electrical power to an electricalload coupled to the receive structure, where the electrical power isused as a power source for the electrical load.
 24. The method of claim21, further comprising impedance matching an electrical load to thereceive structure.
 25. The method of claim 24, further comprisingestablishing a maximum power transfer from the receive structure to theelectrical load.
 26. The method of claim 21, wherein the receivestructure further comprises a magnetic coil.
 27. The method of claim 21,wherein the receive structure further comprises a linear probe.
 28. Themethod of claim 21, wherein the receive structure further comprises atuned resonator coupled to the terrestrial medium.
 29. An apparatus,comprising: a receive structure that receives energy conveyed in a formof a guided surface wave along a surface of a terrestrial medium. 30.The apparatus of claim 29, wherein the receive structure is configuredto load an excitation source coupled to a guided surface waveguide probethat generates the guided surface wave.
 31. The apparatus of claim 29,wherein the energy comprises electrical power, and the receive structureis coupled to an electrical load, and wherein the electrical power isapplied to the electrical load, the electrical power being employed as apower source for the electrical load.
 32. The apparatus of claim 31,wherein the electrical load is impedance matched with the receivestructure.
 33. The apparatus of claim 29, wherein the receive structurefurther comprises a magnetic coil.
 34. The apparatus of claim 29,wherein the receive structure further comprises a linear probe.
 35. Theapparatus of claim 29, wherein the receive structure further comprises atuned resonator.
 36. The apparatus of claim 35, wherein the tunedresonator comprises a series tuned resonator.
 37. The apparatus of claim35, wherein the tuned resonator comprises a parallel tuned resonator.38. The apparatus of claim 35, wherein the tuned resonator comprises adistributed tuned resonator.
 39. A power transmission system,comprising: a guided surface waveguide probe that transmits electricalenergy in a form of a guided surface wave along a surface of aterrestrial medium; and a receive structure that receives the electricalenergy.
 40. The power transmission system of claim 39, wherein thereceive structure loads the guided surface waveguide probe.
 41. Thepower transmission system of claim 39, wherein an electrical load iscoupled to the receive structure and the electrical energy is used as apower source for the electrical load.
 42. The power transmission systemof claim 41, wherein the electrical load is impedance matched to thereceive structure.
 43. The power transmission system of claim 41,wherein a maximum power transfer is established from the receivestructure to the electrical load.